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The Mechanical Response and Parametric Optimization of Ankle-The Mechanical Response and Parametric Optimization of Ankle-
Foot Devices Foot Devices
Kevin Smith University of Central Florida
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STARS Citation STARS Citation Smith, Kevin, "The Mechanical Response and Parametric Optimization of Ankle-Foot Devices" (2016). Electronic Theses and Dissertations, 2004-2019. 5144. https://stars.library.ucf.edu/etd/5144
THE MECHANICAL RESPONSE AND PARAMETRIC OPTIMIZATION OF ANKLE-FOOT
DEVICES
by
KEVIN CHRISTOPHER SMITH
B.S.M.E University of Central Florida, 2016
A thesis submitted in partial fulfillment of the requirements
for the degree of Master of Science
in the Department of Mechanical and Aerospace Engineering
in the College of Engineering and Computer Science
at the University of Central Florida
Orlando, Florida
Summer Term
2016
ii
ABSTRACT
To improve the mobility of lower limb amputees, many modern prosthetic ankle-foot
devices utilize a so called energy storing and return (ESAR) design. This allows for elastically
stored energy to be returned to the gait cycle as forward propulsion. While ESAR type feet have
been well accepted by the prosthetic community, the design and selection of a prosthetic device
for a specific individual is often based on clinical feedback rather than engineering design. This
is due to an incomplete understanding of the role of prosthetic design characteristics (e.g.
stiffness, roll-over shape, etc.) have on the gait pattern of an individual. Therefore, the focus of
this work has been to establish a better understanding of the design characteristics of existing
prosthetic devices through mechanical testing and the development of a prototype prosthetic foot
that has been numerically optimized for a specific gait pattern. The component stiffness, viscous
properties, and energy return of commonly prescribed carbon fiber ESAR type feet were
evaluated through compression testing with digital image correlation at select loading angles
following the idealized gait from the ISO 22675 standard for fatigue testing. A representative
model was developed to predict the stress within each of the tested components during loading
and to optimize the design for a target loading response through parametric finite element
analysis. This design optimization approach, along with rapid prototyping technologies, will
allow clinicians to better identify the role the design characteristics of the foot have on an
amputee’s biomechanics during future gait analysis.
iii
ACKNOWLEDGMENTS
I would like to thank my mentor and advisor Dr. Ali P. Gordon for his support and
encouragement during my academic endeavors. I am additionally grateful for the support all of
the academic faculty members and fellow researchers have offered me help during my time at
the University of Central Florida. I would like to thank my defense committee members, Dr.
Yuanli Bai, Dr. Alain Kassab, and Dr. Patrick Pabian for their time, guidance, and helpful
insights. I would especially like to thank Dr. Bai and Dr. Seetha Raghavan for their support and
use of their equipment.
I also wish to sincerely thank Dr. Les Kramer from TaiLor Made and Stan Patterson from
the Prosthetics & Orthotics Associates for their time, expertise, and contributions that have made
this research possible.
I would additionally like to thank the L.E.A.R.N program and the Research and
Mentoring Program at the University of Central Florida for the academic opportunities and
funding they have provided.
Finally, a special thanks to all of the members of the Mechanics of Materials Research
Group who have helped me along the way.
iv
TABLE OF CONTENTS
LIST OF FIGURES ....................................................................................................................... vi
LIST OF TABLES .......................................................................................................................... x
CHAPTER 1: INTRODUCTION ................................................................................................... 1
CHAPTER 2: LITERATURE REVIEW ........................................................................................ 3
2.1 Prosthetic Design Criteria and Approaches .......................................................................... 3
2.2 Existing Designs .................................................................................................................... 7
2.3 Biomechanics of Prosthetics ............................................................................................... 10
2.4 Experimental Mechanics of Prosthetics .............................................................................. 12
2.4.1 Energy Return ............................................................................................................... 13
2.4.2 Stiffness ........................................................................................................................ 15
2.4.3 Viscoelasticity .............................................................................................................. 18
2.4.4 Fatigue and Proof Testing............................................................................................. 25
2.5 Numerical Approaches Used in Prosthetic Design ............................................................. 27
CHAPTER 3: EXPERIMENTAL APPROACHES ..................................................................... 30
3.1 Load Platform Design and Fabrication ............................................................................... 30
3.2 Preliminary Testing ............................................................................................................. 31
3.3 Compressive Testing ........................................................................................................... 38
3.4 Force Relaxation Testing .................................................................................................... 41
v
3.5 Digital Image Correlation.................................................................................................... 42
CHAPTER 4: EXPERIMENTAL RESULTS .............................................................................. 47
4.1 Compressive Test Results ................................................................................................... 47
4.2 Force Relaxation Results ..................................................................................................... 52
4.3 Digital Image Correlation.................................................................................................... 59
CHAPTER 5: FINITE ELEMENT ANALYSIS .......................................................................... 61
5.1 Renegade Foot Model ......................................................................................................... 61
5.2 Parametric 2D Model .......................................................................................................... 72
5.3 Parametric 3D Model .......................................................................................................... 80
CHAPTER 6: CONCLUSIONS ................................................................................................... 93
6.1 Concluding Statements ........................................................................................................ 93
6.2 Recommendations for future work ...................................................................................... 95
APPENDIX A: CODES ................................................................................................................ 97
APPENDIX B: EXPERIMENTAL DATA ................................................................................ 107
APPENDIX C: SCHEMATICS OF COMPONENTS ............................................................... 113
APPENDIX D: TEST MATRIX ................................................................................................ 116
REFERENCES ........................................................................................................................... 120
vi
LIST OF FIGURES
Figure 1: Freedom Innovations’ Renegade prosthetic foot (Freedom-Innovations, 2015) ............ 2
Figure 2: TLM Prosthetics’ TaiLor Made foot (Prosthetics, 2015) ................................................ 2
Figure 3: Gait Analysis (De Koster, 2004) ..................................................................................... 3
Figure 4: Movements of the ankle ("Dorsiflexion and Plantarflexion,") ........................................ 4
Figure 5: Roll-over shape of different designs prosthetic feet ........................................................ 5
Figure 6: Walking reaction loads (Perry, 1992) .............................................................................. 6
Figure 7: ISO 22675 cyclic loads and angle over time (International Organization for
Standardization, 2006b) .................................................................................................................. 7
Figure 8: SACH foot design ("Postoperative SACH foot (cutaway),") .......................................... 8
Figure 9: Flexible keel dynamic response foot (Kapp et al.) .......................................................... 8
Figure 10: Carbon fiber Freedom Innovations Silhouette® (Innovations) ..................................... 9
Figure 11: a) Single Axis Truper College ("Truper," 2012) b) Multiaxial TruStep College Park
(Thompson, 2006) ......................................................................................................................... 10
Figure 12: Generalized hysteresis loop ......................................................................................... 13
Figure 13: Normalized stiffness at critical load angles during stance phase (Haberman, 2008) .. 17
Figure 14: Standard Linear Model (Reddy, 2013) ........................................................................ 20
Figure 15: Mechanical diagram of Maxwell-Wiechert (a) and Burgers model (b) (Reddy, 2013)
....................................................................................................................................................... 24
Figure 16: (a) Wazau fixed angle test frame (2013) (b) prosthetic test frame with roll-over based
on ISO 22675 (Western) ............................................................................................................... 26
Figure 17: MTS 5kN Insight with custom testing platform .......................................................... 30
Figure 18: Hysteresis loop for 20° Renegade with and without foam cover ................................ 32
vii
Figure 19: Displacement rate test on Renegade foot with cosmetic cover ................................... 33
Figure 20: Load test on Renegade foot with cosmetic cover ........................................................ 34
Figure 21: Force Relaxation of Renegade foot with cover at 20 degrees ..................................... 35
Figure 22: Force-relaxation models for 20° Renegade foot and foam cover ................................ 37
Figure 23: Test loads with regression of idealized gait from ISO 22675 ..................................... 40
Figure 24: Diagram of experimental setup used for Digital Image Correlation (Pan et al., 2014)43
Figure 25: Camera setup with prosthetic foot in load frame ........................................................ 46
Figure 26: Displacement profile of Renegade and TaiLor Made prosthetic foot with and without
foam .............................................................................................................................................. 48
Figure 27: Tangential stiffness profile of Renegade and TaiLor Made foot with and without foam
....................................................................................................................................................... 49
Figure 28: Component stiffness profile of Renegade and TaiLor Made foot with and without
foam .............................................................................................................................................. 50
Figure 29: Energy return profile for Renegade prosthetic foot with and without foam ............... 51
Figure 30: Energy return profile for TaiLor Made prosthetic foot with and without foam .......... 51
Figure 31: Renegade foot time exponent (𝑚) with and without the foam cover ......................... 53
Figure 32: TaiLor Made time exponent (𝑚) with and without the foam cover ........................... 54
Figure 33: Renegade and TaiLor Made foot force exponent (𝑛) with and without the foam cover
....................................................................................................................................................... 55
Figure 34: Renegade foot force relaxation coefficient (𝐴) with and without the foam cover ..... 56
Figure 35: Viscoelasticity parameter of the Renegade and TaiLor Made foot ............................. 57
Figure 36: Time parameter of the Renegade and TaiLor Made foot ............................................ 58
Figure 37: Energy dissipation during force relaxation ................................................................. 59
viii
Figure 38: DIC Von Mises strain in Renegade toe during 20° loading ........................................ 60
Figure 39: Representative CAD model (a) and Renegade foot (b) (Freedom-Innovations, 2015)62
Figure 40: Renegade foot mesh and boundary conditions ............................................................ 63
Figure 41: Response plot of vertical displacement, Young’s modulus, and Poisson’s ratio ........ 64
Figure 42: Von Mises stress in isotropic Renegade foot at -15° loading ...................................... 65
Figure 43: Von Mises stress in isotropic Renegade foot at 20° loading ....................................... 66
Figure 44: Maximum principle (tensile) stresses during loading of toe of the Renegade model . 67
Figure 45: Minimum principle (compressive) stresses during loading of toe of the Renegade
model............................................................................................................................................. 67
Figure 46: Von Mises strain in toe of Renegade isotropic model at 20° ...................................... 70
Figure 47: Parametric foot 2D profile ........................................................................................... 73
Figure 48: Parametric foot 2D mesh and boundary conditions .................................................... 74
Figure 49: FEA Roll over shape ................................................................................................... 77
Figure 50: Anthropomorphic diagram (Drillis et al., 1966) .......................................................... 78
Figure 51: Roll-over trade-off plot of center position (X0) and radius (R) for 2D model ............ 79
Figure 52: Trade off plot of maximum Von Mises stress versus mass in 2D parametric analysis80
Figure 53: Depth (𝐷) dimensions of the foot that were selected for parametric analysis ............ 81
Figure 54: Boundary conditions used on 3D parametric model with 0° toe out ........................... 83
Figure 55: FE model of prosthetic foot with 7° lateral rotation about pylon axis ........................ 84
Figure 56: Trade off plot of tensile a) and compressive stress b) versus mass in 3D parametric
analysis with 0° toe-out ................................................................................................................. 86
Figure 57: Trade off plot of tensile a) and compressive stress b) versus mass in 3D parametric
analysis with 7° toe-out ................................................................................................................. 87
ix
Figure 58: Roll-over tradeoff plot of center position (x0) and radius (R) for 3D model at 0°
toe-out ........................................................................................................................................... 89
Figure 59: Roll-over tradeoff plot of center position (x0) and radius (R) for 3D model at 7°
toe-out ........................................................................................................................................... 89
Figure 60: Work done by vertical-displacement during heel-strike and push-off at 0° toe-out ... 91
Figure 61: Work done by vertical-displacement during heel-strike and push-off at 7° toe-out ... 91
Figure 60: Drawing of baseplate to interface with load frames .................................................. 114
Figure 61: CAD of experimental set-up in MTS 5kN Insight load frame .................................. 115
x
LIST OF TABLES
Table 1: Summary of RehabTech energy return results (Rihs et al., 2001).................................. 15
Table 2: ISO 22675 cyclic load points .......................................................................................... 38
Table 3: Energy return efficiency with and without a cosmetic cover ......................................... 52
Table 4: Comparison of finite element model deflection and Renegade experimental data ........ 64
Table 5: Dimensions of toe and heel neck in bending calculations .............................................. 69
Table 6: Analytical and numerical stress calculations .................................................................. 69
Table 7: Orthotropic Young’s Modulus ........................................................................................ 71
Table 8: Orthotropic Shear Modulus and Poisson’s ratio ............................................................. 71
Table 9: Comparison of stress and deflection ............................................................................... 72
Table 10: Maximum principle stresses due to lateral rotation in original geometry .................... 84
1
CHAPTER 1: INTRODUCTION
In the United States there is currently an estimated 2 million people living with an
amputation and this number expected to reach 3.6 million by 2050. Currently, the leading cause
of amputation in the U.S are dysvascular diseases that originate from underlying conditions such
as diabetes mellitus which account for 54% amputations and 97% of lower limb amputations
(Dillingham et al., 2002; Ziegler-Graham et al., 2008). To improve rehabilitation techniques and
assitive technologies for this growing population of lower limb amputees, a significant body of
literature has been devoloped in the field of biomechanics and medical design with the goal of
restoring function and independence to an amputee. To this end, investigations on lower limb
prosthetic device have identified several key design parameters including, the roll-over shape
(Hansen, Childress, et al., 2004), the viscous behavior (Geil, 2002), the component stiffness, and
the elastic energy return during push-off (Hafner et al., 2002b). However, many gait analysis
studies that examine the performance of existing prosthetic devices fail to report these response
parameters making it difficult to interpt their influence on an specific amputee. This is largely
due to the lack of standardization in reporting the mechanical respone of prosthetic devices.
While several researchers have independently examined the effects of the roll-over shape (Klodd
et al., 2010) and the stiffness (Fey et al., 2011) these prototype designs do not fully report all of
the response parameters. To improve the mechanical characterization and design process of
lower limb prosthetics this paper will consist of two parts. First mechanical testing will be
conducted to evaluate the stiffness, energy return, and viscous properties of a used, but in good
condition Freedom Innovations Renegade foot (Figure 1), as well as, a prototype foot called the
TaiLor Made that was developed by TLM Prosthetics and allows for the clinicians to
2
independently select the stiffness of the toe, heel, and three internal compressive springs in base
chamber shown in Figure 2 .
Figure 1: Freedom Innovations’ Renegade prosthetic foot (Freedom-Innovations, 2015)
Figure 2: TLM Prosthetics’ TaiLor Made foot (Prosthetics, 2015)
Finally this paper will offer an approach to parametrically generate prosthetic feet with a
designed mechanical response using finite element simulations in ANSYS. This optimization
process will allow for faster product development and potentially lead to additional insight on the
influence the prosthetic device has on the biomechanics of amputee.
3
CHAPTER 2: LITERATURE REVIEW
2.1 Prosthetic Design Criteria and Approaches
In the design of prosthetic devices, it is necessary to first develop an understanding of
basic anatomical terms, gait dynamics, and the coupled relationship between the mechanical
stresses on the device and the biomechanical forces on the amputee.
The gait cycle during walking consist of two main phases, the stance and swing phase as
illustrated in Figure 3. The stance phase begins with a heel strike, or initial contact, and
continues forward with the rollover of the ankle until the toe pushes off ground and the leg enters
the swing phase of the gait cycle (Loudon et al., 2008). A gait analysis, which consist of the
measurement the duration of the stance and swing phases, the stride lengths, walking velocity,
and step cadence are commonly used in biomechanics to evaluate the functional performance of
a prosthetic design and will be discussed in greater detail in Section 2.2.
Figure 3: Gait Analysis (De Koster, 2004)
4
During the stance phase, many feet are designed to allow a degree of deflection when placed
under the cyclic loads. This allows for the foot to simulate the dorsiflexion and plantarflexion
motions (shown in Figure 4) of the ankle joint that the foot undergoes during perambulation.
Figure 4: Movements of the ankle ("Dorsiflexion and Plantarflexion,")
The kinematic behavior of this deflection is often described by the roll-over shape of the foot
(Hansen et al., 2014). This is the shape of a cam roller necessary to simulate the path of the ankle
joint based on the mechanical deflections of the prosthesis. Studies by Hansen have suggested
that increasing the roll-over radius can reduce the heel strike on the sound limb and increase the
step length, however, it is important to remember that the roll-over shape is only one of several
important variables in prosthetic design. As Hansen points out, the shape of the foot is
independent from the mechanical behavior of the prosthetic. This is demonstrated in Figure 5
where Hansen has depicted two types of feet from his study with significantly different
mechanical properties, but similar roll-over shapes. The shape of this rollover curve is generated
by plotting the effective center of pressure acting on the foot into the shank axis as the foot rolls
forward (Hansen et al., 2006).
5
Figure 5: Roll-over shape of different designs prosthetic feet
Research in gait analysis (Winter, 2009) and bipedal dynamics (McGeer, 1990) has shown that in
non-amputees the roll-over radius is typically about 15% of an individuals’ height or 30% of
their leg length; and that this radius is often invariant to changes in walking speeds (Hansen,
Childress, et al., 2004) and heel height (Hansen & Dudley, 2004). Additionally, there is
empirical evidence suggesting that there is an optimal roll-over radius and position of the center
of curvature at which metabolic expenditure can be minimalized (Adamczyk et al., 2006).
The expected loads that the prosthetic foot will encounter are often approximated with
reaction forces recorded with a load plate during a gait analysis study. The typically reaction
forces during walking are shown in Figure 6 with respect to body mass. In these plots, the first
peak in the vertical ground reaction force plot represents the heel strike which transfers into the
maximal loading of the heel. The final peak in the vertical gait response represents the toe push-
off. In the anterior-posterior reaction force plot, the first peak represents the breaking force
provided during the loading response of the stance phase while the second peak represents push-
off force opposite to the direction of walking. Finally, the medial-lateral reaction represents
transverse force that occurs as the center of mass shifts side to side.
6
Figure 6: Walking reaction loads (Perry, 1992)
These loads are often further idealized as a symmetric waveform, shown in Figure 7, during
fatigue testing with the ISO 22675 and 10328 standards for the testing of foot and ankle units
using a load frame that supports dynamic roll-over of the ankle (International Organization for
Standardization, 2006a, 2006b). In this plot, the vertical forces and tilt plate angle are
represented as a function of time during the stance phase for the fatigue testing of a foot designed
for a 60kg (P3), 80kg (P4), and 100kg (P5) individual.
Mid-stance
Push-off
Heel-strike
Loading
7
Figure 7: ISO 22675 cyclic loads and angle over time (International Organization for
Standardization, 2006b)
2.2 Existing Designs
With the introduction of high performance composites and improved numerical design,
the lower limb prosthetics industry has evolved over the past 30 years, expanding from designs
focused only on functional recovery to performance based designs that allow for amputees to
participate in sports in activities. This diversity among patient specific needs and recovery goals
has led to the development of a wide variety of prosthetic technologies that can generally be
categorized as fixed joint, passive joint, and active joint prosthetics.
The most traditional fixed joint prosthetic foot today is the solid ankle cushion heel
(SACH) foot which consist of a rigid wooden keel with a foam heel (Figure 8) that simulates
plantar flexion during compression (Nobbe Orthopedics). Because of the simplicity of the
8
design, the SACH foot is generally regarded as a light weight, inexpensive foot with low
maintenance that is well suited to low level ambulators (Highsmith, 2009).
.
Figure 8: SACH foot design ("Postoperative SACH foot (cutaway),")
However, because of their rigid design, SACH feet offer a limited range of motion for
dorsiflexion during the late stance phase of the gait. To increase this range of motion, a number
of prosthetic designs have included a flexible keel made from lightweight high performance
polymer matrix composites (PMCs), such as carbon fiber, aramid fibers, and fiber glass as shown
in Figure 9.
Figure 9: Flexible keel dynamic response foot (Kapp et al.)
9
Among the most successful of the flexible PMC fixed joint prosthetic designs are the
energy storing and return (ESAR) prosthetic feet which consist of an elastic leaf spring (Figure
10) design that allows for a for a greater degree of ankle flexibility and dynamic energy return
during push-off than traditional wood and foam prosthetics (Hafner et al., 2002a). Because of
their flexibility, high energy return, and low maintenance, ESAR feet are particularly well suited
to athletics sporting events and activities.
Figure 10: Carbon fiber Freedom Innovations Silhouette® (Innovations)
Feet with passive joints, such as single axis (Figure 11a) and multi-axis feet (Figure 11b)
offer amputees a greater range of motion that leads to improved stability on uneven terrain as
shown in (Nobbe Orthopedics). This is due to the frictionless bearing in the ankle joint allows
foot to come into contact with ground quicker simulating flexion in single axis feet and eversion
or inversion in multi-axis feet. While single axis feet are viable option for all amputees, this
increase increased stability is particularly important to transfemoral amputees. However, as the
designs become increasingly complex, the patient and prosthetist must balance the cost, the
weight of the device, and the accommodation period to learn to walk again (Highsmith, 2009).
10
a)
b)
Figure 11: a) Single Axis Truper College ("Truper," 2012) b) Multiaxial TruStep College Park
(Thompson, 2006)
2.3 Biomechanics of Prosthetics
Since the introduction of the first carbon fiber ESAR foot in the early 1980s, there have
been numerous studies on prosthetic feet to determine which types of designs work well and the
design criteria that makes them successful. To accomplish this goal, research studies typically
focus on either the patient-prosthetic interaction using a biomechanical analysis, or an unbiased
structural analysis by applying static, dynamic and cyclic loads to the prosthetic device.
During a biomechanical analysis researchers often focus on at least one of the six areas of
a gait analysis including: kinematics, kinetics, muscle activation, metabolic expenditure, and the
stride and temporal characteristics (Hafner et al., 2002a). While each of these areas of study does
provide insight into the performance of the prosthetic, these studies often provide results are
often statistically insignificant and at times conflicting with both each other, and the perception
of the patient.
11
Hafner summarized the results of a number of these studies which showed gait analysis
with ESAR foot can have self-selected walking velocity anywhere from 0.00-13.11% higher than
a traditional prosthetic foot (Barr et al., 1992; Lehmann, Price, Boswell-Bessette, Dralle, &
Questad, 1993; Lehmann, Price, Boswell-Bessette, Dralle, Questad, et al., 1993; Macfarlane et
al., 1991; Nielsen et al., 1988; Powers et al., 1994; Snyder et al., 1995; Torburn et al., 1990).
While the increase in walking velocities in most studies are statistical insignificant (P > 0.05), all
of the reviewed studies on ESAR feet have shown some increase in walking velocity with a
mean value of 4.7% (Hafner et al., 2002a). This reflects the positive patient feedback received in
surveys and polls performed Menard and others (Menard et al., 1989; Romo, 1999). However,
the clinical significance of these results has often been questioned as the day-to-day self-selected
walking velocity can vary in patients as much as 7.1% of the mean (Kadaba MP et al., 1989;
Perry, 1992).
Further kinematic analysis has revealed that the cadence (steps / minute) remains
relatively unchanged between SACH feet and ESAR feet (Powers et al., 1994; Snyder et al.,
1995). Instead the increase in walking velocity is largely attributed to an increase in stride length
due to the increase in the range of motion in the flexible ankle; where stride length consist of a
forward step with each leg (Hafner et al., 2002a). Although not all authors go into detail, Barr
and Macfarlane have suggested that the increased stride length is due to an increase in the step of
the sound leg as there is a delay in the unloading of the bodyweight off of the prosthetic foot
(Barr et al., 1992; Macfarlane et al., 1991).
In examining the effects of energy return on the kinetics during walking, researchers
found that there was an increase in the posterior anterior force (Lehmann, Price, Boswell-
Bessette, Dralle, Questad, et al., 1993; Powers et al., 1994). This was due to the plantarflexion of
12
the foot as elastic energy stored by the leaf spring design was released. Interestingly researchers
also found that there was often a significant decrease in vertical heel reaction force on the sound
side, but only minimal reduction in vertical forces on the affected side (Lehmann, Price,
Boswell-Bessette, Dralle, Questad, et al., 1993; Powers et al., 1994; Snyder et al., 1995). These
results support the previously mentioned conclusions from Barr and Macfarlane, the increased
flexibility of the toe allows for a reduction in force during the two support phase of the
subsequent step (Barr et al., 1992; Macfarlane et al., 1991).
Metabolic expenditure experiments have consistently shown that amputees demonstrate
an elevated heart rate and 55% to 83% higher oxygen consumption when walking at similar
speeds as non-amputees (Hoffman et al., 1997; Waters et al., 1999). The goal of many advance
prosthetics is to minimize the additional exertion placed on the patient, however, only a few
metabolic test have shown improvement using ESAR feet when compared to traditional feet. In a
study by Nielsen that examined oxygen update and heart rate during walking, it was shown that
the ESAR feet perform are less demanding at higher speeds, but the difference was less
noticeable at speeds below 2.5mph (Nielsen et al., 1988). Nevertheless, there was no significant
difference seen in muscle activation between ESAR feet and SACH feet (Torburn et al., 1990).
Despite the lack of statistically significant biomechanical data to support improvements
during walking, there is an apparent trend that the energy that is returned by the ESAR toe during
the push off phase of the gait cycle does assist the amputee.
2.4 Experimental Mechanics of Prosthetics
As previously discussed, biomechanical studies are subjected to a great deal of variance
due small sample sizes and large variability between amputees. To provide a more repeatable
13
evaluation of the prosthetics, bench top testing is conducted to determine the mechanical strength
and behavior of a prosthetic foot. These tests can include both fatigue and proof testing of the
prosthesis often following the ISO 22523 and 22675 standards, as well as characterization of the
mechanical properties such as stiffness, force relaxation, roll-over shape, and the energy return of
dynamic feet.
2.4.1 Energy Return
The energy return of an ESAR prosthetic foot is a measure of the elastic energy released
by the foot relative to the potential energy stored during loading. During bench top testing, the
cyclic work is measured through integration of the force displacement Eq.(2.4.1), which is
represented by the hysteresis loop shown in Figure 12. Here the energy loss as a result of internal
friction is represented by the area within the hysteresis loop (Hafner et al., 2002b).
1
0
x
xW F dx (2.4.1)
Figure 12: Generalized hysteresis loop
14
This energy loss is often expressed as a percentage shown in Eq.(2.4.2), with the energy returned
during unloading relative to the work during loading.
0
1
1
0
x
unloadx
return x
loadx
F dx
F dx
(2.4.2)
Although energy return testing has yet to be standardized in literature, the American
Orthotic Prosthetic Association (AOPA) has provided guidelines on the energy testing of
prosthetic feet to help classify prosthetic feet. According to the AOPA a dynamic response foot
is classified as a foot that has an efficiency of at least 75% on the toe and 82% efficiency on the
heel (American Orthotic Prosthetic Association, 2010). In review of a study conducted by the
Rehabilitation Technology Research Unit at Monash University on 12 different model feet
shown in Table 1, it is noted that in general feet made from high performance composites tend to
perform with higher efficiency.
15
Table 1: Summary of RehabTech energy return results (Rihs et al., 2001)
Foot Type Keel Energy
Return % Keel Material
Human Foot 119.6 -
Seattle Foot M+IND, Natural SNF150 70.7 Delrin
Flex Foot Modular II 61.7 Carbon Fiber
Quantum VESSA Truestep 25 type A N1562 59.5 Fiber Glass
SAFE Cambell Childs 58 Urethane
Multi Axis Foot Blatchford Multiflex Foot 509153-67 56.7 Carbon Fiber
Dynamic Foot - Otto Bock ID10 39.2 Timber
Carbon Copy II 35.2
Kevlar /
Nylon
SACH Foot Otto Bock IS70 32.1 Timber
Greissinger Foot Otto Bock 1A13 31.7 Timber
SACH Foot Kingsley Wayfarer (Post Op. Flatfoot) K10 19.1 Timber
SACH Foot Otto Bock IS51 15.9 Timber
STEN Kingsley 15.8 Timber
While the energy return efficiency is the most commonly reported data statistic in literature,
according to Geil the actual amount of energy return to the late stance phase is far more
significant. It was observed in Geil’s study on dynamic response model feet that the most
compliant foot tested, the College Park’s TruStep, may have had the lowest efficiency at 68.7%,
but the prosthetic foot provided the greatest amount of energy return at 7.10 J during unloading
(Geil, 2001).
2.4.2 Stiffness
The mechanical stiffness and viscous properties of the prosthetic foot play a fundamental
role in the absorption of energy and gait kinematics for amputees. This is due to the relatively
large elastic deformations that the ESAR feet undergo during the stance phase of the gait.
However, as Geil points out, the selection of optimal prosthetic device for the individual patient
16
often presents a challenge for prosthetists and clinicians as the mechanical behavior of the
prosthetic foot is not standardized. When properties such as stiffness are reported they often use
a subjective scale that varies between manufacturers. To help clinicians to make better-informed
decisions regarding the selection of a prosthetic device, independent studies have helped to
provide additional information on the mechanical behavior select devices and their performance
during gait analysis (Geil, 2001).
Mechanical testing of the stiffness of prosthetic feet is typically conducted by placing a
representative load on both the heel and toe of the foot and measuring the deflection of the foot.
Two types of tests used to determine the mechanical behavior are uniaxial loading in a controlled
load frame and impact testing with a weighted pendulum. In a study by Geil, a universal testing
frame was used to determine that the stiffness of nine models of prosthetic feet which ranged
from 28-76 N/mm when 800N compressive load was applied to the foot when it was 12° in
plantarflexion (Geil, 2001). In a study by Klute, impact testing was conducted to determine the
behavior of the heel of seven models of prosthetic feet when angled at 20° in dorsiflexion with
kinetic energy similar to high load rates experienced during running; which can range from
151.9 - 213.9% of an individuals body weight per second in non-amputees (Logan, 2007). It was
found that the heel stiffness of several types of prosthetic feet ranged from 27-68 N/mm (Klute et
al., 2004).
While the individual testing of the heel and the toe provides characterization of the foot
during the heel-strike and push-off, these test do not describe the behavior during the mid-stance
phase of the gait. To characterize the behavior of the prosthetic device over the entire gait cycle,
Haberman conducted uniaxial testing at fifteen data points over an idealize load curve that was
adopted from the ISO 22675 standard. While Habermans’ tests found similar toe stiffness as Geil
17
at 12°, Haberman also found that the stiffness of several prosthetic feet varied over the duration
of the stance phase and exhibited a peak in stiffness when both the heel and the toe are providing
support at 0° or 300ms based on the ISO 22675 loading curves. This change in compliance
during roll-over is attributed to the contact mechanics and the non-uniform geometry (Haberman,
2008).
Figure 13: Normalized stiffness at critical load angles during stance phase (Haberman, 2008)
To better understand the role of the prosthetic foot stiffness on walking, Fey conducted a
series of gait analysis using three variations of a 3D-printed thermoplastic prosthetic foot with
levels of mechanical stiffness (Fey et al., 2011). The nominal level of the three feet was matched
to a Freedom Innovations Highlander foot through mechanical testing and finite element
modeling; the other two feet were designed to be 50% more compliant, and 50% stiffer than the
nominal foot (South et al., 2009). Fey found that a more compliant foot allows for an increased
ankle flexibility and prosthetic energy storage in the mid-to-late stance phase, however, a greater
amount of muscle activity in the both residual limb and sound limb was also observed (Fey et al.,
2011). This suggests that there is an apparent trade-off between energy return and stability.
18
2.4.3 Viscoelasticity
While the influence of the viscous properties of an ankle-foot device on an amputee’s gait
pattern are poorly understood, Klute has suggested that increasing the energy dispersed by the
viscoelastic behavior during heel loading could potentially reduce the risk of skin and soft tissue
damage in an amputee’s residual limb (Klute et al., 2004). To this end, several methods have
been presented in literature including the use of impact and quasi-static loading to quantify the
viscoelastic behavior of prosthetic feet. In Klute’s approach, impact testing was conducted at a
20° angle on the heel of seven commonly prescribed prosthetic feet within their cosmetic foam
covers and within walking shoes, running shoes, and orthotic shoes. The percentage of energy
dissipation during impact was calculated by dividing the work energy from deformation during
impact force by the kinetic energy of a weighted impactor pendulum, i.e.
2
100%1
2
s
FdxD
mv
(2.4.3)
Where 𝑥 is the deformation of the heel, F is the force during impact, 𝑚 is the mass of the
pendulum, and 𝑣 is the velocity of the pendulum. It was observed that the energy dissipation
percentage for the tested prosthetic feet ranged from 33.6% to 52.6%. Klute noted that wearing a
shoes significantly increased the energy dissipation during impact. In the case of Trulifes’
dynamic response Seattle foot, the dissipation energy was increased from 45.3% by itself, to
63.0% in walking shoes, 73.0% in running shoes, and 82.4% in orthotic shoes (Klute et al.,
2004).
19
To quantify the viscous behavior, Klute developed a constitutive model of the force
during impact with a nonlinear spring in parallel with a directionally-dependent damper as
shown in Eq.(2.4.4),
, signeb dF x x ax x cx x (2.4.4)
Here the coefficients (a) and exponent (b) represent the nonlinear hardening behavior during
elastic deformation and the coefficient (c), and exponents (d) and (e), represent the directionally
dependent energy dissipation. The sign (�̇�) term goes to 1 when �̇� is positive, -1 when �̇� is
negative, and 0 when �̇� is equal to zero. The coefficients in this model are found through
numerical regression, however, due to the non-linear behavior it is difficult to directly compare
the coefficients values between different model feet. Klute points this out by examining the
impact coefficients of a SACH foot and the VariFlex foot, one of several ESAR feet tested. It
was found that the (a) coefficient VariFlex foot was higher than the SACH feet, suggesting that
it would have a higher peak load. However, this is not the case due to a greater position
dependent exponent (b) in the VariFlex foot, which instead led to a smaller peak load in the
VariFlex than the SACH foot (Klute et al., 2004).
Other researchers have taken a more traditional approach to evaluating the viscous
properties of ankle-foot devices by conducting stress relaxation, creep, and load rate testing. To
adapt these mechanics of material experiments for the component level, Geil developed an
angled fixture in a universal load frame to apply uniaxial loads to a prosthetic device with a 12
degree angle against a fixed flat surface (Geil, 2002). This allowed Geil to apply a constant load
to the prosthetic device and measure the deformation over time during creep testing, to apply a
20
fixed displacement and measure the decay in force over time during stress relaxation testing, and
measure the force response during tests conducted at a constant strain rate. To model the viscous
behavior of nine different types of ESAR prosthetic feet under these loading conditions, Geil
applied the three parameter standard linear viscoelasticity material model which consists of an
elastic spring element (𝑘1) that is in series with an a Voigt element that is composed of elastic
spring (𝑘2) that is parallel to a rate dependent viscous damper (𝜂); the standard linear model can
be expressed in its mechanical form shown in Figure 14.
Figure 14: Standard Linear Model (Reddy, 2013)
This spring and dashpot representation can be converted to its equation form by applying
compatibility conditions such that the stress (𝜎) and strain (𝜖) components of the parallel
elements must equal the total stress and strain in the system, i.e.
1 2 (2.4.5)
1 2 (2.4.6)
Additionally, the stress-strain relations of the two spring elements can be determined by Hooke’s
law, the series spring and dashpot can be determined from the linear viscosity,
21
1 2 2k (2.4.7)
2 2 (2.4.8)
1 1k (2.4.9)
Combining the compatibility conditions and the stress-strain relationships leads to a first order
differential equation that is capable of modeling both stress relaxation and creep strain as
follows,
22
1 1
1kd d
kk dt k dt
(2.4.10)
When solving Eq.(2.4.10) for conditions of constant strain (𝜖0), it becomes apparent that the
isolated spring 𝑘1 element allows modeling of the long duration stiffness of the component,
while the parallel spring 𝑘2 and its dashpot 𝜂 capture the transient behavior of the material, i.e.
21 20
t exp
kk k
(2.4.11)
Similarly, Eq. (2.4.10) can be solved for the creep response of a material with constant stress and
�̇� equal to 0 as shown in Eq.(2.4.12),
2 1 2
1 1
0
22 1
11 exp
k tk k
k k k k k
(2.4.12)
22
As well as the constant strain rate (𝜖̇) response as shown in Eq.(2.4.13)
21 1 exp
tkk
(2.4.13)
While the standard linear model is typically used for engineering stress (𝜎), strain (𝜖), and strain
rate (𝜖̇) at the material level, these values could not be measured directly due to the geometry of
the foot. Instead, Geil made an approximation of these values at the component level in order to
quantify the viscoelastic behavior of the device. The component level stress response was
estimated by dividing the force over the foots’ plantar surface area, and the strain response was
derived by dividing the deformation distance between the foot pylon and the loading surface by
the original un-deformed distance. The viscoelasticity coefficients for the constitutive model for
the three types of loading were determined from a quasi-Newton optimization routine of the
experimental data for each foot. These experiments by Geil support previous findings by
Lehmann that the Flex foot is more compliant than the Seattle foot, which is more compliant
than a SACH foot (Lehmann, Price, Boswell-Bessette, Dralle, & Questad, 1993). While Geil
admits there were some inaccuracies between the standard linear model and the experimental
data, it was noted that the series spring constant (𝑘2) in the Flex foot was the highest out of all
the feet tested, thus leading to a greater initial stiffness. This suggests that in the design and
selection process of a prosthetic foot, there may be additional insight in examining viscoelastic
behavior of the ankle-foot device rather than the device stiffness alone (Geil, 2002).
Due to the complex behavior between the ankle-foot devices, the cosmetic foam cover,
and orthotic shoes the standard linear model may not be sufficiently robust to capture the viscous
23
response for prosthetic devices (Geil, 2002). To fill this gap in knowledge, this thesis will further
examine the viscous response of prosthetic devices using the Norton-Bailey power law, the
Maxwell-Wiechert model, and the Burgers model by replacing the stress parameter (𝜎) and
strain parameter (𝜖), respectively with force (𝐹) vertical displacement (𝛿). In the case of the
Norton-Bailey power law it follows that,
n m n mA t AF t (2.4.14)
Where, by analogy the stress exponent (𝑛) can referred to as the force exponent, and the strain
hardening coefficient (𝐴) can be referred to as the force coefficient. Additionally, 𝑡 represents
time and 𝑚 represents the time exponent (Betten, 2008). During stress relaxation conditions
Eq.(2.4.14) can be re-written as follows,
0
1/n
mF
At
(2.4.15)
Here the force relaxation response is determined from the constant displacement (𝛿0) and the
viscous parameters. An additional power law often used in the constitutive modeling of rubbers
(Larson, 1985) is shown in Eq.(2.4.16),
0
mF F t (2.4.16)
24
It is noted that the Norton-Bailey law from Eq.(2.4.15) reduces to the form of Eq.(2.4.16), when
the force exponent goes to -1. In such cases, the material could be considered to have a linear
stress response.
Akin to the standard linear model, the Maxwell-Wiechart and Burgers model are derived
from a spring and dashpot diagram shown in Figure 15. The Maxwell-Wiechert, or generalized
Maxwell model, consists of a single elastic spring element (𝑘𝑒) along with an arbitrary number
( 𝐽 ) of Maxwell spring-dashpot elements to fully describe the stress relaxation behavior (Reddy,
2013).
a)
b)
Figure 15: Mechanical diagram of Maxwell-Wiechert (a) and Burgers model (b) (Reddy, 2013)
Following a similar conversion of stress and strain as in Eq.(2.4.14), the Maxwell-
Wiechert model is expressed as follows,
11
1 1
t exp
J
e
j
kF k k
(2.4.17)
25
Where the viscoelastic spring (𝑘) and dashpot (𝜂) elements (Reddy, 2013). Due to the structure
of Eq.(2.4.17), the Maxwell-Wiechert equation allows for a high degree of accuracy for stress
relaxation with a sufficient number of Maxwell elements. On the other hand, the Burgers model
shown in Eq.(2.4.18),
1
2
1 2 2 1
1 11 exp
tk tF
k k
(2.4.18)
consists of only four spring and dashpot element; however, because of the equation structure it is
particularly well suited to complex creep flow (Reddy, 2013).
2.4.4 Fatigue and Proof Testing
Prosthetic feet are regularly subjected to cyclic loading during the day-to-day
perambulation of an amputee. Over time, this cyclic loading leads to void formation and crack
growth through various failure mechanisms in composite laminate feet including de-bonding,
delamination, and fatigue failure of the fiber reinforcement. To predict the lifespan of the device,
many manufactures subject their lower limb prosthetic components to fatigue and proof testing at
the peak loads and critical angles observed on the heel and toe of the foot during the stance phase
of a gait analysis.
In literature, the fatigue testing of a prosthetic foot is often conducted by applying a load
at a critical angle with one or more linear actuators (Figure 16a) or through the use of highly
specialized testing equipment (Figure 16b) that allows for the dynamic rollover of the prosthetic
26
foot (Toh et al., 1993; Unnthorsson et al., 2008). While the inclusion of dynamic roll-over during
of prosthetics is considered to a better representation of the service conditions for the device,
relatively few publications have been made using fatigue testing with this full range of motion
due cost and complexity (Daher, 1975; Wevers et al., 1987). Instead fatigue testing of lower limb
prosthetics is typically conducted with the use of uniaxial testing frames which allows similar
life cycle predictions as dynamic roll-over testing. This approach was used by Toh who observed
that the foam heel is common site of failure in SACH feet often beginning to breaking down
after 6 months of wear or 10,000 cycles of fatigue testing (Toh et al., 1993).
(a)
(b)
Figure 16: (a) Wazau fixed angle test frame (2013) (b) prosthetic test frame with roll-over based
on ISO 22675 (Western)
In recent years, many manufactures utilized the ISO 22675 and ISO 10328 standards for
the structural testing of lower limb prosthetic devices and their components. As previously
discussed in section 2.1, these two standards provide recommended test loads for fatigue and
structural testing based on an idealized gait loads for a designed body mass. To comply with
27
these standards, prosthetic feet must endure the following fatigue and static loads without crack
formation:
Cyclic loading of the gait force shown in Figure 7 for 2 × 106 cycles, followed by a proof
test at 175% of the peak gait load for 30 seconds
Static ultimate test equal to 350% of the peak load in the gait cycle
However, a displacement criterion for permanent or loaded deformations has been also been used
as a failure criterion (Toh et al., 1993; Unnthorsson et al., 2008). While components high
performance composites, such as carbon fiber, tend to exhibit good fatigue resistance, this is not
the case with lower cost polymer and foam prosthetic devices used in developing countries.
Independent testing has shown that many SACH feet and thermoplastic monolimb prosthetic
devices have a service life less than 100,000 cycles or approximately one year of use (Jensen et
al., 2000; Jensen et al., 2007; Toh et al., 1993).
2.5 Numerical Approaches Used in Prosthetic Design
To help address the multivariable challenge in the design of lower limb prosthetics, the use of
finite element analysis (FEA), optimization routines, and recent 3D printing techniques have
been used to generate more robust designs, develop a better understanding of the patient-
prosthesis adaptation, and improve the lifespan of the device.
In a study by Fey the influence of prosthetic foot stiffness on walking was examined through
the use of three variations of a thermoplastic prototype prosthetic foot, generated through FEA
modeling and selective laser sintering (SLS) (South et al., 2009). The initial findings from this
biomechanical study showed that a more compliant foot leads to increased propulsive energy, it
28
also required greater muscle activation to compensate for decreased stability (Fey et al., 2011).
To further verify these findings in a separate study, Fey was further able to use FE modeling to
examine the muscle activation for walking with a 2D dynamics simulation of the lower body and
prosthesis (Fey et al., 2013).
Finite element modeling has also been used in the optimization of the composite material of a
prosthetic device. This approach has been used by Limmer who conducted experimental and
finite element analysis of the elastically stored strain energy density for comparing the effects of
the orientation of woven T300 carbon fiber ply-lay ups and 3D woven composites that utilize a
designed elastic modulus (Limmer et al., 1996). In doing so, Limmer found that the localized
control of the stiffness in the prosthetics’ material allowed for increased elastically stored energy
at both natural and fast walking speeds.
Due to the complex shape of a prosthetic device, finite element modeling has also allowed
for the generation of life and reliability models without the need to conduct comprehensive
component level fatigue testing. In a study by Chen the gait reaction forces from a 55 year old
patient were used to identify the fatigue life and probability of failure of the patients
polypropylene monolimb prosthetic over the service life of the device (Chen et al., 2006). This
was accomplished by statistically modeling the material life, the uncertainty of the two peak
loads during walking, and the cumulative damage on the device. According to Chen, the material
life of thermoplastics is well represented by Wirschings S-N curve model in Eq.(2.5.1) where the
cycles to failure (𝑁𝑓) is determined from the fatigue stress (𝑆), the fatigue strength exponent
(𝑚) and the fatigue strength coefficient (𝐾).
m
fN K S (2.5.1)
29
The logarithmic uncertainty (𝐵) of the estimate peak loads (𝑆𝑖𝑒) during walking substituted into
Miner’s rule, as shown in Eq. (2.5.2), for linear cumulative damage where the median life of the
material (𝑁𝑖) and number of cycles (𝑛) is used to predict the total damage (𝛼) incurred on the
material.
1 2
1 2
m m
m e eB S S nn n
N N K
(2.5.2)
This cumulative damage expression was then integrated in the lognormal probability distribution
function with experimentally determined uncertainty values for peak loads and the fatigue
strength coefficient to determine the probability to failure over 𝑛 cycles during the service of the
device.
30
CHAPTER 3: EXPERIMENTAL APPROACHES
3.1 Load Platform Design and Fabrication
To characterize the mechanical behavior of commercially available prosthetic feet, a
custom testing rig was designed and fabricated with the ability to interface with a both a 5kN
MTS Insight and 50kN Instron unaxial load frame allowing for the uniaxial compression testing
of prosthetic feet at the representative loads and angles of the stance phase of the gait cycle. The
testing rig, shown in Figure 17, consisted of a rigid flat plate that was held at the target angle by
a sine vise with stack blocks that were clamped to the 5kN MTS load frame that was used during
the benchtop testing of the prosthetic feet.
Figure 17: MTS 5kN Insight with custom testing platform
31
To mount the machine clamps to the load frame an aluminum interface plate was developed with
its dimensions shown in the Appendix C. The rigid T-shaped load plate held by sine vise
consisted of 1 x 12 x 6 in. steel that was welded to a 1 inch bar. This setup allowed for the
examination of the force-displacement history during cyclic loading of the foot.
3.2 Preliminary Testing
Prior to the mechanical testing of the prosthetic feet, preliminary cyclic compressive tests
were conducted to determine the effects of the loading rate, the load magnitude, and
preconditioning effect on the stiffness and energy return efficiency. Additionally, preliminary
force relaxation tests were conducted to determine the minimum test duration necessary to fully
describe the viscous relaxation of the component. These preliminary tests were conducted on a
size 28 Freedom Innovations Renegade prosthetic foot with a size 28 foam cover with a
prescribed foot stiffness of seven, which is recommended for body mass up to 100 kg under high
impact conditions, or a 130 kg individual under low impact applications (Freedom-Innovations,
2015). Following the ISO 22675 recommended loads and angles for a P5 category foot, the
prosthetic was monotonically loaded to the target force using a 5kN MTS Insight and the custom
testing rig.
Despite behaving elastically under the cyclic loads, the force displacement of the foot took
on a nonlinear profile due to the foot geometry and contact mechanics. A typical force
displacement response for the prosthetic foot with and without the foam cover is shown in Figure
18. The linear stiffness (𝐾) of the prosthetic foot during the preliminary testing was measured by
calculating the tangential slope of the final 80% of the force displacement plot. The work energy
lost during loading and unloading, shown by the area within each hysteresis loop, was calculated
32
by integration of the force (F) and displacement (𝛿) with Eq.(2.4.1) using the trapezoidal
integration method. It was noted that the Renegade foot was highly efficient in contrast to the
other of the market prosthetic described in Table 1. At the 20° angle the Renegade foot returned
98% of the elastically stored energy during unloading and 92% of the stored energy with the
cosmetic foam cover.
Figure 18: Hysteresis loop for 20° Renegade with and without foam cover
The preliminary displacement rate controlled test on the Renegade prosthetic foot was
conducted at a 15 degree incline to a target load of 1271N with a controlled crosshead
displacement speed between 0.2 mm/s and 10 mm/s as shown in Figure 19. It was noted that the
displacement rate, and consequently the load rate, had minimal effect on the stiffness and energy
return of the prosthetic foot during loading. On average, at the 15° angle the Renegade foot in its
foam cover demonstrated a stiffness of 68.70 N/mm during compressive loading and returned
89.2% of the elastically stored work energy (W) during unloading. These results suggest that the
0
200
400
600
800
1000
1200
1400
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
Forc
e, F
(N)
Displacement, δ (m)
w/o cover
with cover
20° angle at 2.25mm/s
33
ISO 22675 standard’s recommendations of a loading rate between 100-250 N/s for static proof
testing is also an acceptable guideline for evaluating the energy return and the stiffness of
prosthetic feet.
Figure 19: Displacement rate test on Renegade foot with cosmetic cover
The effect of the force magnitude on the stiffness and energy return of the Renegade
prosthetic was examined with mechanical loading of the foot against a 15 degree incline load
plate at a constant displacement rate of 2 mm/s. The results of these test are shown in Figure 20.
As expected, the amount of work energy absorbed by the prosthetic foot is linearly proportional
to the force placed on the foot. However, it was also observed that there was both an increase in
the energy loss and an increase in the stiffness as the load was increased. These trends are
attributed to the geometry of the foot during bending. This supports the assumption that
evaluation of the stiffness and energy return must be conducted at the expected loads and angles
during service.
0 100 200 300 400 500 600 700 800
0
10
20
30
40
50
60
70
80
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Load Rate, Ḟ (N/s)
Stif
fnes
s, K
(N/m
m)
Wo
rk, W
(J)
Displacement rate, δ̇ (mm/s)
Absorb
Release
Stiffness15° angle
34
Figure 20: Load test on Renegade foot with cosmetic cover
To reflect the behavior during prolonged use, preliminary tests were conducted to determine
the amount preconditioning necessary to remove any viscous effects that could be appear during
the mechanical testing. This tests consisted of cyclically loading the Renegade prosthetic foot in
its foam cover at a 15 degree angle to a load of 1271 N for ten cycles. It was determined that the
energy return had stabilized by the third cycle. It was decided that a five cycle preload would be
appropriate to eliminate any viscous effects present during the mechanical testing. The viscous
effects of the prosthetic foot were examined by loading the Renegade foot with and without its
foam cover at a rate of 0.2 mm/s to a load of 1276 N at an angle of 20 degrees and held at this
load for a duration of 180s (Figure 21) based similar testing conducted by Haberman (Haberman,
2008). It was noted that there was only a 0.02% difference in load between 170s and 180s;
therefore it is assumed that a 180s test duration provided sufficient time for the viscous effects to
fully decay. During the preliminary relaxation test of the Renegade foot it was noted that the foot
with its cover experienced 2.6% decay in force, whereas the foot without the cover experienced
0
10
20
30
40
50
60
70
80
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
0.0 200.0 400.0 600.0 800.0 1000.0 1200.0 1400.0
Stif
fnes
s (N
/mm
)
Wo
rk (
J)
Load (N)
Absorb
Release
Stiffness
15° angle at 2 mm/s
35
1.25% decay in force. This indicates that the cosmetic foam cover does influence the viscous
behavior of the prosthetic foot, however, the short term viscoelastic effects are relatively
negligible. For a 100 kg individual during prolonged walking with the Renegade foot and its
cover, the maximum viscoelastic deformations would be approximately 0.85 mm.
Figure 21: Force Relaxation of Renegade foot with cover at 20 degrees
To characterize the force relaxation data, the standard linear viscoelasticity model, the
Norton-Bailey, the Burgers model, and the generalized Maxwell-Wiechert model, that were
previously discussed in Section 2.4.3 , were fitted to the experimental data using numerical
regression software. For convince, these models are reproduced as follows,
Power Law: 0
mF F t (3.2.1)
Norton-Bailey:
1/n
mF
At
(3.2.2)
0.97
0.975
0.98
0.985
0.99
0.995
1
1.005
0 20 40 60 80 100 120 140 160 180
No
rmal
ized
Fo
rce
(N)
Time, t (s)
20° without cover
20° with cover
36
Standard linear: 21 2
t exp
kF k k
(3.2.3)
Burgers:
1
2
1 1 2 2
1 11 exp
tktF
k k
(3.2.4)
Maxwell-Wiechert: 2 1
22
1
t exp
N
n
kF k k
(3.2.5)
To adapt the viscoelasticity models in Eq.(3.2.3) through Eq. (3.2.5) for the force relaxation
testing, the 𝑘1 stiffness, which represents the long duration elastic properties, was set equal to the
experimental force 𝐹(𝑡), at time (t) equals 180 seconds, over the constant displacement (𝛿0) as
shown in Eq.(3.2.6).
0
1
180F tk
(3.2.6)
The long duration stiffness 𝑘1 represents the force behavior after force has completely decayed
and viscosity has become negligible. A comparison of these three models of these models with
the preliminary force relaxation data shown in Figure 22.
37
Figure 22: Force-relaxation models for 20° Renegade foot and foam cover
While the Norton-Baily model goes to infinity at time equal to zero, it was observed that for all
other time increments the 3-term Norton- Bailey model produced the best fit of the force
relaxation data coefficient of determination (𝑅2) of 0.999. Both the power law and Maxwell-
Wiechert model were also able to produce a similarly accurate model with 𝑅2 equal to 0.999 and
0.998 respectively, however, in both cases, the residual error was greater than the Norton-Bailey
model. In contrast, the standard linear and Burgers model did not fit as well with a correlation
coefficient of 0.92 and 0.78, respectively. It was also noted than the time exponent (𝑚) in the
power law (-0.0062) and the Norton-Bailey equation (-0.0068) were similarly valued. This is a
result of the force exponent (𝑛) having a value of -1.45; as 𝑛 goes towards -1, both the power
law and the Norton-Baily equation would have an identical time exponent.
1230
1240
1250
1260
1270
1280
1290
0 50 100 150
Forc
e (N
)
Time, t (s)
Data
Burgers Eq.
Standard Linear Eq.
Maxwell-Wiechert Eq.
Norton-Baily Eq.
Power Law Eq.
38
3.3 Compressive Testing
The forces and angles used in the mechanical characterization of the Renegade foot and
the TaylorMade prototype foot were based on the idealized gait load curve from the ISO 22675
standard shown in Figure 7. The tabulated values of this loading curve, shown in Table 2, was
used to develop an analytical model of the idealized gait response for the normalized vertical
loads and the angle of the tibia with respect to the vertical. Fifteen data points over this idealize
loading pattern were then selected for mechanical testing in the testing rig.
Table 2: ISO 22675 cyclic load points
Time, t (ms) Angle (deg.) P5 Load (N) P4 Load (N) P3 Load (N)
0 -20 0 0 0
30 -19.5 331 306 238
60 -19 663 612 477
90 -18 996 919 716
120 -16.5 1221 1126 878
150 -15 1273 1173 915
180 -13 1215 1120 873
210 -10.5 1092 1007 785
240 -7.5 969 893 697
270 -4 880 811 632
300 0 850 783 611
330 4 879 810 632
360 8 966 891 694
390 12 1086 1003 781
420 16 1204 1110 866
450 20 1256 1158 903
480 24 1198 1105 861
510 28 971 895 698
540 32 643 593 463
570 36 321 296 231
600 40 0 0 0
39
The idealized vertical loads 𝐿(𝑡) and loading angles 𝐷(𝑡) were developed using a three-term
sinusoidal equation.
( ) cos(7.1308 cos(sin(5.2553 0.003431 )))L t t (3.3.1)
2( ) 6.4034 19.9878 cos( 0.004274 )D t t t (3.3.2)
The constants in Eq.(3.3.1) and Eq.(3.3.2) were developed through numerical regression of the
normalized loads from the tabular data in Table 2. During mechanical testing at the P3, P4, and
P5 load levels, the force over time was determined by multiplying Eq.(3.3.1) by the peak load in
each respective category. Fifteen data points over this load pattern were selected at every five
degrees from -20 to 40 degrees, with an additional two data points at -19.25 and -17.5 degrees to
capture the slower initial roll-over during heel loading. The tabulated ISO 22675 data, the data
points for mechanical testing, and the analytical models of vertical force and angle are shown in
Figure 23.
40
Figure 23: Test loads with regression of idealized gait from ISO 22675
The test points are reasonably distributed to capture the behavior of the foot over the entire
loading pattern. Alternatively the six-term polynomial provided in the ISO standard could also
have been used to generate the idealized loading pattern. To examine the influence of the foam
cover on the stiffness and energy return of the prosthetic feet, mechanical testing was
additionally conducted on the feet at the three critical angles of -15, 0, and 20 degrees shown in
red on Figure 23. A test matrix for the appropriate load and the desired testing angles for each
foot was generated based on the recommended body mass and ISO 22675 testing standard. This
matrix is listed in Appendix D.
During the preliminary compressive testing, the stiffness (𝐾) was measured from the
tangential slope of the linear region of the force displacement plot. This is a typical approach
used to evaluate the elastic modulus during material testing. However it was noted in
-30
-20
-10
0
10
20
30
40
50
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 100 200 300 400 500 600
An
gle,
θ(d
eg)
No
rmal
ized
fo
rce,
F(N
/N)
Time, t (ms)
ISO 22675 Loads
Test Loads
Test w/o cover
L(t)
ISO 22675 Angle
Test Angles
D(t)
41
Figure 18 that the force-displacement response of the prosthetic foot has a nonlinear elastic
behavior. For this reason, it was determined that the overall component stiffness (𝐾𝑐) should also
be evaluated at each angle during the compressive testing.
maxc
max
FK
(3.3.3)
Where the component stiffness is equal to the peak force (𝐹𝑚𝑎𝑥) over the maximum deflection
(𝛿𝑚𝑎𝑥)of the prosthetic foot. While the linear stiffness provides a well understood analogy to the
elastic modulus of a material, the overall component stiffness may be more beneficial for
clinicians.
Additionally, the energy return at each of the fifteen test angles was calculated from the
force-displacement plots following a similar procedure as the preliminary testing. The work
energy during loading and unloading of at each test angle was calculated through the integration
of Eq.(2.4.1) via the trapezoidal method. Following Eq.(2.4.2), this allowed for the calculation of
the work energy efficiency of the foot at each test angle.
3.4 Force Relaxation Testing
During the relaxation testing of the Renegade and TaiLor Made prosthetic feet, each foot
was subjected to a cyclic preconditioning followed by force relaxation testing at fifteen angles
with the cover and three critical angles without the foam cover. Based on the preliminary testing
in section 3.3, the preconditioning consisted of loading prosthetic foot from rest to the target
force five times. The force relaxation testing of the prosthetic feet consisted of loading the
prosthetic foot to the target load and then holding the MTS crosshead in a fixed position for a
42
duration of 180s while recording the decay of force on the load cell. The fifteen angles used for
testing with the foam cover and foot consisted of the data points shown in Figure 23, while the
relaxation testing without the foam cover was conducted three critical angles of -15, 0, and 20
degrees.
Based on the preliminary testing in section 3.2, the force relaxation of each angle was
characterized through the use of the Norton-Bailey creep model shown in Eq.(3.4.1).
0
1/n
mF
At
(3.4.1)
By fitting the force coefficient (𝐴), the force exponent (𝑛), and the time exponent (𝑚) to the
decay in force (𝐹) over time (𝑡) through numerical regression.
3.5 Digital Image Correlation
During the structural testing of the prosthetic feet the crosshead displacement was used to
measure the vertical deflection of the foot; however, because of the complex geometry of the
component, this displacement provides little insight into the localized stress and strain acting on
the component. To identify these localized values, digital image correlation (DIC) was
performed during mechanical testing to optically measure the surface strain on the carbon fiber
layers during the structural loading of the Renegade and TaiLor Made prosthetic feet. While
optical strain measurement techniques, such as digital image correlation, are primarily used to
develop a displacement map of 2D and 3D specimen surfaces during material testing, it has
previously been shown by Parnell that these techniques can also be used to evaluate prosthetic
feet at the component level during mechanical proof testing (Parnell, 2014).
43
The digital image correlation technique is an expansion of the laser speckle
interferometry method that determines the displacement field of a speckle painted surface as a
specimen is subjected to mechanical loads. In contrast to traditional laser interferometry methods
that require highly specialized equipment, DIC can be conducted with an ordinary monotone
light source, a computer, and the charge-coupled device (CCD) array found in a high resolution
digital video camera as depicted in Figure 24 (Pan et al., 2014). Following a similar approach as
Parnell, the DIC analysis was conducted during by painting the side of each foot with white paint
flecks to provide to provide a sufficient contrast to allow optical measurements of the strain
through the ply thickness.
Figure 24: Diagram of experimental setup used for Digital Image Correlation (Pan et al., 2014)
As the prosthetic foot was subjected to mechanical loads, the speckles of the painted
surface experience relative motion due to the surface deformations of the foot. This movement is
recorded in a series of images showing a progression of the surface from its original state, to its
final deformed state. These images are digitally converted to an intensity map by the CCD-array
allowing for the displacement mapping of target features between successive images. The target
point 𝑃(𝑥, 𝑦) in a reference state was mapped to the point 𝑃∗(�̃�, �̃�) in the image of the deformed
44
state using the displacement components u and v as shown in Eq.(3.5.1) and Eq. (3.5.2) (Dally et
al., 2005).
,x x u x y (3.5.1)
,y x v x y (3.5.2)
The displacement mapping parameters of components u and v are expressed as a 12 parameter
second order Taylor series expansion of an arbitrary point 𝑆𝑝 = 𝑃(𝑥0, 𝑦0) that exists in a subset
of points S in the image, as follows:
2 2 2
2 2
0 0 2 2
1 1Δ Δy Δ Δ Δ Δ
2 2
u u u u ux x u x x y x y
x y x y x y
(3.5.3)
2 2 2
2 2
0 0 2 2
1 1Δ Δy Δ Δ Δ Δ
2 2
u u u u uy x u x x y x y
x y x y x y
(3.5.4)
These mapping parameters are determined by minimizing a correlation coefficient C, shown in
Eq.(3.5.5), between the target point locations in reference state and its location in the deformed
state. In this equation, the gray scale pixel intensity of the reference state and the deformed state
are represented using continuous bi-cubic spline functions respectively shown in Eq.(3.5.6) and
Eq.(3.5.7). The pixel intensity function 𝑔(𝑥, 𝑦) of the reference state depends only on the pixel
location in the reference image and its spline coefficients 𝑎𝑚𝑛, whereas the intensity function
ℎ(�̃�, �̃�, 𝑃) of the deformed state depends on the new pixel location, the coefficients 𝑏𝑚𝑛, and the
45
mapping parameters P which represents both the displacement vector as well as the coefficient 𝛼
to adjust any changes in the brightness between the two images (Dally & Riley, 2005).
2
2
,
p
p
p p
S S
p
S S
g S h S P
C
g S
(3.5.5)
3 3
0 0
, m n
mn
m n
g x y a x y
(3.5.6)
3 3
0 0
, , m n
mn
m n
h x y P b x y
(3.5.7)
To record the surface strain of the prosthetic feet during loading a CCD camera with a
2448x2048 resolution and Tokina AT-X Pro D M100 F2.8 lens was centered on the ankle joint
of the prosthetic foot. Due to space limitations, it was necessary to conduct the DIC compressive
testing on a 50 kN MTS Insight Wide load frame with a as shown in Figure 25. The DIC analysis
of the recorded images was conducted using Vic – 2D version 2009 software.
46
Figure 25: Camera setup with prosthetic foot in load frame
47
CHAPTER 4: EXPERIMENTAL RESULTS
4.1 Compressive Test Results
Uniaxial compression test were conducted on the Freedom Innovations Renegade foot
and the TaiLor Made foot to determine the deflection, the stiffness, and energy return during
quasi-static loading at select fifteen selected angles over the idealized loading pattern with and
without their cosmetic covers as depicted in Figure 23. The magnitude of the loading pattern for
each respective foot was determined by the manufactures body mass recommendations. The
Renegade foot examined in this study was size 28 foot with a stiffness of 7 and recommended for
body between 100kg and 130kg, which corresponds to P5 loading conditions by the ISO 22675
standard (Freedom-Innovations, 2015). The size 23 TaiLor Made foot that was examined was
recommended for a 60kg individual under the P3 loading conditions. The TaiLor Made
prosthetic foot was also equipped with three blue internal tibial springs. The cosmetic covers
used in this test were both Freedom Innovations covers as the TaiLor Made covers were not yet
commercially available. A direct comparison of the displacement for each foot over the loading
curve is shown in Figure 26. The normalized vertical loads are also plotted as a reference for heel
strike and toe push-off.
48
Figure 26: Displacement profile of Renegade and TaiLor Made prosthetic foot with and without
foam
Interestingly, both feet exhibited an identical amount of deflection during heel strike without
their foam covers and at their respective loads. However, as roll-over progressed the TaiLor
Made foot was subjected to greater deflections than the Renegade foot. It was additionally noted
there was relatively large amount of the heel deflection in the TaiLor Made foot with its cosmetic
cover. This is attributed to an imperfect fit between the cosmetic cover and the TaiLor Made
foot.
Both the linear stiffness and the component stiffness of each test angle were calculated
following the procedures described in Section 3.3. The slope of the linear region of the force
displacement plot is shown in Figure 27 and the component stiffness from Eq.(3.3.3) is examined
0
0.2
0.4
0.6
0.8
1
1.2
0.00
10.00
20.00
30.00
40.00
50.00
60.00
0 100 200 300 400 500 600
No
rmal
ized
ver
tica
l lo
ad
Dis
pla
cem
ent,
δ(m
m)
Time, t (ms)
Renegade w/ Cover Renegade w/o Cover TailorMade w/ Cover
TailorMade w/o Cover Vertical Load
49
in Figure 28. By comparing these two plots it becomes apparent that linear stiffness may
overestimate the behavior of the device. This is due to the nonlinear hardening behavior that
compressive loading. For this reason, it is assumed that the overall component stiffness may be
more useful to clinicians.
Figure 27: Tangential stiffness profile of Renegade and TaiLor Made foot with and without foam
0
0.2
0.4
0.6
0.8
1
1.2
0
20
40
60
80
100
120
140
160
180
200
0 100 200 300 400 500 600
No
rmal
ized
Fo
rce
(N)
Stif
fnes
s, K
(N
/mm
)
Time, t (ms)
TailorMade stiffness TailorMade stiffness w/o foam Renegade stiffness
Rengade stiffness w/o cover Normalized vertical load
50
Figure 28: Component stiffness profile of Renegade and TaiLor Made foot with and without
foam
Further examination of Figure 28 shows that the cosmetic cover can reduce stiffness behavior of
the prosthetic foot assembly. However, this effect was observed to a much greater extent on the
heel of the TaiLor Made foot. It was also noted at 0° the stiffness of the TaiLor Made was
significantly lower than that of the Renegade foot; a potential result of the tibial springs in the
TaiLor Made foots’ design.
Using the cyclic force displacement data from each of these tests the strain energy density
during loading and unloading was calculated through integration of the work with Eq.(2.4.1) and
the trapezoidal method. The energy return efficiency (𝜂) was calculated as a ratio of the work
energy during loading to the energy during unloading following in Eq.(2.4.2) at each angle over
the load pattern. The energy return profile for the Renegade and TaiLor Made foot are plotted in
Figure 29 and Figure 30 respectively.
0
0.2
0.4
0.6
0.8
1
1.2
0
20
40
60
80
100
120
0 100 200 300 400 500 600
No
rmal
ized
Fo
rce
(N)
Co
mp
on
ent
stif
fnes
s, K
C(N
/mm
)
Time, t (ms)
TailorMade stiffness TailorMade stiffness w/o foam Renegade stiffness
Rengade stiffness w/o cover Normalized vertical load
51
Figure 29: Energy return profile for Renegade prosthetic foot with and without foam
Figure 30: Energy return profile for TaiLor Made prosthetic foot with and without foam
0%
20%
40%
60%
80%
100%
120%
0.000
5.000
10.000
15.000
20.000
25.000
0 100 200 300 400 500 600
Ener
gy R
etu
rn E
ffic
ien
cy,
η(%
)
Wo
rk (
J)
Time, t (ms)
Energy Absorb Energy Release Absorb w/o cover
Release w/o cover Efficiency Efficiency w/o cover
0%
20%
40%
60%
80%
100%
0
2
4
6
8
10
12
14
16
0 100 200 300 400 500 600
Ener
gy R
etu
rn E
ffic
ien
cy,
η(%
)
Wo
rk (
J)
Time (ms)
Absorb Release Absorb w/o cover
Release w/o cover Effiency Efficiency w/o cover
52
Here it is observed that the both the Renegade and the TaiLor Made foot exhibit a much higher
energy return efficiency in comparison to various other prosthetic feet that were listed in Table 1.
Interestingly, the peak energy capacity on the toe of both the Renegade and TaiLor Made foot
did not occur at the at the peak load at 20 degrees, instead it occurred 25 degrees. It was
additionally noted that the return energy efficiency (𝜂𝑐) in the cosmetic cover decreased by 5%
and 6% in comparison to the efficiency (𝜂) without a cover on the heel and toe of the Renegade
foot. In contrast, the TaiLor Made prosthetic foot experienced a 15% and 3% decrease in
efficiency on its heel and toe, however, the energy return efficiency of the TaiLor Made actually
increased by 12% with its cosmetic cover at vertical loading as listed in Table 3.
Table 3: Energy return efficiency with and without a cosmetic cover
Time (ms) Angle Efficiency w/o cover (𝜼) Efficiency with cover (𝜼𝒄)
Renegade
144 -15 99% 93%
299 0 88% 85%
451 20 98% 92%
TaiLor
Made
144 -15 88% 74%
299 0 59% 71%
451 20 83% 80%
This unexpected behavior is likely due to the internal tibial springs of the TailorMade foot falling
into proper alignment when the cosmetic cover is present to help distribute the force of the load
plate.
4.2 Force Relaxation Results
Following the force relaxation procedure developed in preliminary testing section 3.2 and
the experimental outline in section 3.4 force relaxation testing was conducted on the Renegade
and TaiLor Made foot at fifteen different angles with its cosmetic cover and an additional three
53
angles without its cosmetic cover. The force relaxation over time was fitted to the Norton-Bailey
power law to determine the time exponent, the force exponent, and the force coefficient. When
the time exponent (𝑚) of the Renegade foot is plotted against over the duration of roll over as
shown in Figure 31, it is observed that without the foam cover the value of 𝑚 remains relatively
constant, however, with the foam cover the value of 𝑚 varies as the foot progresses through the
roll-over.
Figure 31: Renegade foot time exponent (𝑚) with and without the foam cover
This behavior is due to the time dependent behavior of the cosmetic cover. As the foot
progresses through the roll-over, the foam experiences different levels of localized stress leading
to variations in the viscous behavior of the overall component. Conversely, when the carbon
fiber prosthetic is tested without the cosmetic cover, the prosthetic device is less dependent on
time with 𝑚 closer to the value of zero. Similarly, when plotting 𝑚 values of the TaiLor Made
0
0.2
0.4
0.6
0.8
1
1.2
-0.018
-0.016
-0.014
-0.012
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
0 100 200 300 400 500 600
No
rmal
ized
load
, F
Tim
e Ex
po
nen
t (m
)
Time (ms)
with foam without foam Normalized Load
54
prosthetic foot, in Figure 32, it is noted that the heel region of the prosthetic foot exhibited a
much more negative time exponent. This indicates that a greater amount of energy is dissipated
by the heel of the TaiLor Made foot within its cosmetic cover.
Figure 32: TaiLor Made time exponent (𝑚) with and without the foam cover
When the force exponent (𝑛) of Norton-Bailey equation was plotted over the idealized
load cycle as shown in Figure 33, it was noted that the values of 𝑛 were similar in both the
TaiLor Made and Renegade foot, and that over the load cycle 𝑛 shows a similar trend as the
normalized load. This is because the 𝑛 exponent characterizes the loading function while the
force coefficient (𝐴) has a greater correlation with the stiffness of the component.
0
0.2
0.4
0.6
0.8
1
1.2
-0.050
-0.045
-0.040
-0.035
-0.030
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
0 100 200 300 400 500 600
No
rmal
ized
load
, F
Tim
e Ex
po
nen
t (m
)
Time (ms)
with foam without foam Normalized Load
55
Figure 33: Renegade and TaiLor Made foot force exponent (𝑛) with and without the foam cover
When the force coefficient (𝐴) is plotted by itself in Figure 34 it was observed that the
coefficients of the TaiLor Made foot are relatively constant, while the Renegade foot coefficients
peak near the 0° loads. This is likely a result of the both the heel and the toe coming in to contact
with the load plate and the additional compliance of the tibial springs in the TaiLor Made foot.
0
0.2
0.4
0.6
0.8
1
1.2
-2.00
-1.90
-1.80
-1.70
-1.60
-1.50
-1.40
-1.30
-1.20
-1.10
-1.00
0 100 200 300 400 500 600
No
rmal
ized
load
, F(N
/N)
Forc
e Ex
po
nen
t (n
)
Time (ms)
Renegade with foam Renegade w/o foam TailorMade with foam
TailorMade w/o foam Normalized Load
56
Figure 34: Renegade foot force relaxation coefficient (𝐴) with and without the foam cover
The Norton-Bailey power law from Eq.(2.4.15) can also be rewritten as shown in
Eq.(4.2.1) where 𝐴−1/𝑛 can be considered a load parameter and m/n can be considered a viscous
time parameter for decay in force over time (𝑡) at fixed displacement (𝛿0).
1/ /
0
n m nF A t (4.2.1)
Interestingly, when this load parameter is plotted over the load cycle, as shown in Figure 35, it is
observed that there both the TaiLor Made and Renegade prosthetic foot behavior similarly in the
heel region, despite testing the TaiLor Made and Renegade foot at different force values.
0
0.2
0.4
0.6
0.8
1
1.2
0
500
1000
1500
2000
2500
3000
3500
0.00 100.00 200.00 300.00 400.00 500.00 600.00
No
rmal
ized
Lo
ad, F
(N
/N)
Forc
e C
oef
fici
ent
(A)
(N
/m)
Time (ms)
Renegade with foam Renegade w/o foam TailorMade with foam
TailorMade w/o foam Normalized Load
57
Figure 35: Viscoelasticity parameter of the Renegade and TaiLor Made foot
This would suggest that the load parameter 𝐴−1/𝑛 is representative of the loading function and
that it is likely influenced by the compliance of the cover. Similarly, the stress exponent and time
exponent can be plotted together as a time parameter m/n shown in Figure 36.
0
0.2
0.4
0.6
0.8
1
1.2
0
20
40
60
80
100
120
140
160
180
0 100 200 300 400 500 600
No
rmal
ized
Lo
ad, F
(N
/N)
Load
Par
amet
er, A
^(-1
/n)
Time (ms)
Renegade with foam Renegade without foam
TailorMade with foam TailorMade without foam
Normalized Load
58
Figure 36: Time parameter of the Renegade and TaiLor Made foot
In doing so, it becomes easier to see that that the foam cover contributes significantly greater
viscosity to the heel of the TaiLor Made foot. Again, this increased viscoelasticity is likely the
result of the geometry of the heel and the fit of the cover. This is further demonstrated by Figure
37, in which the energy dissipated over the duration of the relaxation test is plotted for the
TaiLor Made and Renegade foot.
0
0.2
0.4
0.6
0.8
1
1.2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 100 200 300 400 500 600
No
rmal
ized
Lo
ad, F
(N
/N)
Tim
e P
aram
eter
, m/n
Time (ms)
Renegade with foam Renegade without foam
TailorMade with foam TailorMade without foam
Normalized Load
59
Figure 37: Energy dissipation during force relaxation
While Klute (Klute et al., 2004) had previously suggested that analysis of the viscous properties
of a prosthetic foot may provide additional insight in to their ability to dissipate energy, the
findings in Figure 37 suggest this capacity is predominately caused by foot covers and shoes; and
that by themselves the carbon fiber prosthetics may dissipate energy similarly.
4.3 Digital Image Correlation
Optical strain measurements were taken on the prosthetic feet during loading at 20
degrees using the digital image correlation (DIC) technique with a CCD camera and the Vic 2D
imaging software as described in Section 3.5. During this process a contour map was generated
of the surface Von Mises strain in the Renegade foot during loading at 20 degrees as shown in
Figure 38. Here it is observed that the maximum Von Mises strain in the geometry occurs at the
inner and outer radius of the toe similar to curved beam in bending.
0
0.2
0.4
0.6
0.8
1
1.2
0
1
2
3
4
5
6
7
0.00 100.00 200.00 300.00 400.00 500.00 600.00
No
rmal
ized
Lo
ad, F
(N
/N)
Ener
gy D
issi
pat
ed
(J)
Time (ms)
Renegade with foam Renegade w/o foam TailorMade with foam
TailorMade w/o foam Normalized Load
60
Figure 38: DIC Von Mises strain in Renegade toe during 20° loading
It was also noted that the software could not pick up on the all of the surface points of the toe
cross-section, however, it can be inferred that the component experiences a surface strain near
the maximum recorded strain value of 0.00452 along the inner and outer radius. Due to time
constraints the optical strain of the TaiLor Made foot will be explored in a future paper.
61
CHAPTER 5: FINITE ELEMENT ANALYSIS
The finite element method (FEM) has been utilized in this study for the evaluation of a
prosthetic foot designs by examining the loading response, the strain energy, the maximum
stress, the reaction moment, and the kinematic profile during roll over. This approach allows for
the development of a conceptual device that meets set performance criteria under the gait
response for a specific individual. For the purpose of this study, the gait response has been
idealized with the forces and angles from the ISO 22675 standard during cyclic loading. To
verify this approach, a Computer-Aided-Design (CAD) prosthetic foot model was developed in
SolidWorks based on the geometry of the Freedom Innovations Renegade prosthetic foot and
subjected to the ISO 22675 loading pattern in the ANSYS Workbench Mechanical FEA
software. An approximation of the material properties for the FEA model were developed by
comparing the experimental response of the Renegade foot in Section 4.1 with the FEA model
through a parametric analysis. To optimize this FEM model for target performance criteria, such
as the strain energy capacity, roll over shape, mass, reaction moment, and factor of safety,
additional parametric analyses were conducted through topographic optimization.
5.1 Renegade Foot Model
The material of the Renegade foot in the FEM analysis was assumed to behave
isotropically and to be fully described by only the elastic modulus and Poisson’s ratio. A series
of parametric static analysis were conducted on the FEA model to determine the approximate
values of these properties by comparing the vertical deflection of the pylon with the experimental
62
data for the Renegade foot during quasi-static loading at -15, 0, and 20 degrees. Additionally, it
was noted during the compressive testing of the Renegade foot that the over load spring did not
come into contact with the heel spring of the foot during loading. For this reason, the inclusion
of the overload spring in the initial model was omitted as shown in Figure 39.
a)
b)
Figure 39: Representative CAD model (a) and Renegade foot (b) (Freedom-Innovations, 2015)
To simulate the behavior of the foot during bending a 20-node hexahedral element was used to
mesh the CAD. While shell elements are often used during the modeling of composites, brick
element were selected for this model to ensure the accurate modeling of the geometry thickness
and the contact behavior. To reduce the computational cost of the model a symmetric boundary
condition was applied in the sagittal plane (XY) of the foot as shown in Figure 40. Additionally,
constraints were added to the model to prevent movement of the pylon in the X-direction and the
separation of the foot’s contact elements with the plate.
Omitted overload spring
63
Figure 40: Renegade foot mesh and boundary conditions
During the parametric analysis, a range of potential composite material properties from
literature were examined (ACP Composites, 2014; Corum, 2001). The Poisson’s ratio (𝜈) was
set to range between 0.1-0.4 and the Young’s Modulus (𝐸) was set to range between 40-100
GPa. A twenty-five data point parametric analysis was developed using the central composite
method within Ansys’ Design Exploration. To match the behavior of the Renegade foot observed
in Section 4.1, a genetic aggregation response surface optimization was used to determine the
material properties necessary to produce a vertical displacement (𝑈𝑌) of -39.60 mm at 20°, a
displacement of -8.71 mm at 0°, and -15.88 mm at -15° in the FEM model. The response plot of
the vertical displacement at 20° with various values of the Young’s modulus and Poisson’s ratio
is shown in Figure 41.
64
Figure 41: Response plot of vertical displacement, Young’s modulus, and Poisson’s ratio
After adding additional verification points to the response surface, it was determined that the
FEM model would require a Young’s modulus of approximately 67 GPa and a Poisson’s ratio of
0.34 in the heel, and modulus of 42.7 GPa and Poisson’s ratio of 0.14 in the toe, as summarized
in Table 4.
Table 4: Comparison of finite element model deflection and Renegade experimental data
Angle, 𝜽 (degrees) FEA Deflection, UY (mm) Actual Deflection, UY (mm)
-15.00 -14.80 -15.88
0.00 -11.94 -8.71
20.00 -38.53 -39.6
These results suggest that the stiffness of carbon fiber used in the Renegade foot is between 42.7
and 67 GPa, which is consistent with 0/90° woven carbon fiber fabric found in literature and the
65
aerospace industry. However, it was noted that the Poisson’s ratio of carbon fiber typically falls
between 0.05 and 0.1 for 0/90° woven composites, depending on the manufacture and fiber type
(ACP Composites, 2014; Corum, 2001). This discrepancy is likely the result of the simplification
of the material model and not including the overload spring of the Renegade foot.
Further examination of the FEA model reveals the maximum equivalent stress in the foot
geometry appears along the inner surface of the fillets in both the heel and the toe as shown in
Figure 42 and Figure 43 respectively at the -15° and 20° loading. In the plot of the 20° loading it
was noted that a singularity occurred at the tip of the toe due the sharp edges of the geometry in
contact with the plate; these unrealistic stress values have been omitted. It was shown that with
the estimated material properties that the foot geometry would experience peak equivalent stress
of 582.6 MPa during heel loading and 357.48 MPa during toe push-off.
Figure 42: Von Mises stress in isotropic Renegade foot at -15° loading
66
Figure 43: Von Mises stress in isotropic Renegade foot at 20° loading
It was also observed that the during loading the curvatures in the Renegade foot behave similar
to a curved beam, with the stress on the maximum principle (tensile) on the outer radius and the
minimum principle (compressive) stress on the inner surface as shown in Figure 44 and Figure
45 respectively.
67
Figure 44: Maximum principle (tensile) stresses during loading of toe of the Renegade model
Figure 45: Minimum principle (compressive) stresses during loading of toe of the Renegade
model
68
To verify the combined loading experienced by the FEA model, the curved beam formulas for
bending stress on the inner radius (𝜎𝐵𝑖) and outer radius (𝜎𝐵𝑜
), shown in Eq.(5.1.1) and
Eq.(5.1.2), and the normal stress equation Eq.(5.1.3), were applied to the neck of the keel and
heel during their respective peak loads (Ugural et al., 2011).
i
i
B
i
M R r
A r R r
(5.1.1)
O
o
B
o
M R r
A r R r
(5.1.2)
/N F A (5.1.3)
Where 𝑟𝑖, 𝑟𝑜, �̅� are respectively the inner, outer, and average radius of the smallest cross-section
in the foot, 𝐹𝑦 represents the vertical compressive force, 𝑀𝑧 is the moment during bending, h
represents the thickness of the foot cross section, b represents the width of the foot, A is the
cross-section area, and R is the radius of the neutral axis in a curved beam which is calculated by
Eq.(5.1.4).
ln /o i
hR
r r (5.1.4)
The analytical stress was calculated using the superposition principle, shown in Eq.(5.1.5), to
combine the normal and bending stress.
C N B (5.1.5)
69
The dimensions used in these calculations were measured from the CAD model shown in Table
5. Additionally, the 𝑀𝑧 experienced during the loads was estimated by multiplying the vertical
load 𝐹𝑦 by the distance from the fillet in the CAD to the contact point.
Table 5: Dimensions of toe and heel neck in bending calculations
Toe (20° load) Heel (-15° load)
Outer radius, 𝑟𝑜 (cm) 3.74 0.87
Inner radius, 𝑟𝑖 (cm) 1.40 0.37
Thickness, h (cm) 1.00 0.50
Width, b (cm) 3.80 5.00
Vertical load, 𝐹𝑦 (N) -1273 -1271
Moment, M (N·m) 426.46 197.0
A comparison of the analytical and numerical stress values, shown in Table 6, suggest that the
FEA model is accurate, however, it is noted that the principle stresses experienced by the heel
exceeds the maximum compressive strength of most composites.
Table 6: Analytical and numerical stress calculations
Toe (20° load) Heel (-15° load)
FEA 𝜎𝑖 (MPa) -378.87 -654.31
FEA 𝜎𝑜(MPa) 280.45 391.77
Analytical 𝜎𝑖 (MPa) -415.20 -654.96
Analytical 𝜎𝑜 (MPa) 282.30 361.87
This would suggest that a CAD error was during the modeling of the dimensions of the heel in
the Renegade foot. To further validate the approximation of the elastic modulus of the material
70
used by the toe, the Von Mises strain of the foot during loading at 20 degrees, shown in Figure
46, was compared to the previously observed strain the DIC shown in Figure 38.
Figure 46: Von Mises strain in toe of Renegade isotropic model at 20°
Here it is observed that the maximum strain in the model occurs at the symmetric boundary
condition in the middle of the foot at a value of 0.0086, and that the free surface of the foot has
strain value of 0.0059 and 0.0061 respectively along the outer and inner radius of the toe. These
values are similar to the maximum strain of 0.0045 recorded by the DIC. This would suggest that
these initial simulations have provided a reasonable an approximation of the Renegade foot
geometry and its’ material properties.
To prepare a more realistic FEA model for parametric analysis, a transversely isotropic
material model was developed from based on materials values found in literature for simulation
of both the heel and the toe. The in plane properties of this constitutive model were developed
71
from Corum’s material data for 0/90° woven Thronel T300 carbon fibers in Baydur 420 IMR
urethane matrix. The out of plane elastic the elastic modulus (𝐸𝑒𝑝𝑜𝑥𝑦) and Poisson’s ratio
(𝜈𝑒𝑝𝑜𝑥𝑦) were developed using the properties of Baydur 426 IMR which bears a similar rating as
Baydur 420 IMR (Covestro, 2015; EpoTek, 2015). These values were additionally used with
Eq.(5.1.6) to approximate the out of plane shear modulus, 𝐺𝑋𝑍 and 𝐺𝑌𝑍 as summarized in Table 7
and Table 8.
2(1 )
epoxy
ij
epoxy
EG
(5.1.6)
Table 7: Orthotropic Young’s Modulus
Direction Young’s Modulus, E (GPa)
X 44.9
Y 44.9
Z 10.5
Table 8: Orthotropic Shear Modulus and Poisson’s ratio
Direction Poisson’s Ratio, ν Shear Modulus, G (GPa)
XY 0.05 2.96
YZ 0.30 4.04
XZ 0.30 4.04
The FEA model with transversely orthotropic properties produced a similar stresses distribution
and vertical deflection as the isotropic model as shown in Table 9. These similarities between the
transversely isotropic and the isotropic FEA model suggest that the 0/90° woven carbon fiber is a
reasonable estimate of the material the prosthetic foot.
72
Table 9: Comparison of stress and deflection
Transversely
Isotropic
Isotropic Experimental
UY (mm) at -15° -10.65 -14.80 -15.88
UY (mm) at 20° -36.48 -38.53 -39.6
Stress 𝜎𝑉𝑀 (mm) at -15° 596.8 582.6 -
Stress 𝜎𝑉𝑀 (mm) at 20° 365.8 357.48 -
Additionally, it is inferred that an isotropic model may be acceptable for modeling the bending
behavior of a curved beam when additional information is unavailable.
5.2 Parametric 2D Model
In the process of developing a parametrically generated design, a two-dimensional model
prosthetic foot model with an idealized 10 cm depth was prepared from profile the Renegade
CAD. While this model does not accurately reflect a finalized geometry for manufacturing, the
use of a two-dimensional model allows for reduced computational solve time and rapid design
exploration of the planar geometry. The initial thickness of the keel (𝑇𝑘1), the tapered thickness
of the keel (𝑇𝑘2), and thickness in the heel (𝑇𝐻) were selected as parametric input variables.
Additionally, to eliminate rebuild errors in the Solidworks CAD, it was necessary to fully
constrain the profile with the length parameters used in the Section 5.1 model as shown in Figure
47; the parametric variables are shown in red, and the constrained dimensions are shown in
black. A transition point from the tapering of the keel thickness from 𝑇𝐾1 to 𝑇𝐾2 is also shown
below the keel radius in the toe.
73
Figure 47: Parametric foot 2D profile
During the design exploration of the possible geometries, it was decided that the 𝑇𝐻 and 𝑇𝐾2
dimensions would be constrained between the values of 0.4 and 0.6 cm, and that the 𝑇𝐾1
dimensions would be constrained between 0.75 and 1.25 cm.
The parametric simulation of this 2D profile was conducted using plane strain element
formulation with a refined quadrilateral mesh along the bends in the foot geometry as shown in
Figure 48. Similar to the FEA model presented in Section 5.1, this 2D simulation examined the
response of the parametric design at several critical angles during static loading. To properly
align the contact surfaces in the simulation, an APDL command snippet was inserted into the
ANSYS Workench model. This code repositioned the model such that the foot nodes come into
contact with the rigid plate elements at any load angle. Additionally, weak spring elements were
attached to each body to prevent rigid body movement; these springs apply a negligible force
74
that does impact the results of the simulation, however, they are used to fully define the model
for a static simulation.
Figure 48: Parametric foot 2D mesh and boundary conditions
Because the orthotropic model was not supported in this 2D solver, an isotropic assumption of
the previously developed constitutive model in Section 5.1 was utilized; where the Poisson’s
ratio was set 0.3 and the Young’s modulus was set to 44.9 GPa.
75
A central composite method was used to generate fifteen variations of the 2D model,
each of which were simulated at four angles of the roll-over. Several goal driven constraints were
applied to parametric results to develop a feasibility plot of the data. These included minimizing
the peak Von Mises stress in the model at -15° (𝜎𝑉𝑀−15) and 20° (𝜎𝑉𝑀20
), minimizing the mass
(m) of the prosthetic design, and maximizing the elastic strain energy (𝑊20) during push off at
20°, and achieving a targeted roll-over radius; which was calculated by transforming the center
of pressure center during roll-over in the global axis (𝐶𝑂𝑃𝑋𝑌) into the shank or tibial axis
(𝐶𝑂𝑃𝑠). To find the 𝐶𝑂𝑃𝑋𝑌 in the FEA model, the weighted average position (𝑋𝑎𝑣𝑔) of the
contact nodes (𝑁) was calculated with Eq.(5.2.1),
1
1
i
i
N
y iiavg N
y ii
F XX
F
(5.2.1)
Where 𝐹𝑌𝑖 represents the vertical forces on each node (i) and 𝑋𝑖 represents their position in the
global X-axis. Because the center of pressure always occurs at the global Y-axis location of the
contact plate (𝑌𝑝𝑙𝑎𝑡𝑒), the 𝐶𝑂𝑃𝑋𝑌 can be determined as follows,
,XY avg plateCOP X Y (5.2.2)
To transform the COPS into the shank coordinate system, a key-point was generated in the global
coordinate system at the 𝐶𝑂𝑃𝑋𝑌 location. The position of the key-point was then exported in a
local coordinate system aligned with the tibia. To perform these calculations in the FEA model,
an APDL command snippet was inserted into the ANSYS Workbench model. This code is
76
shown in Appendix A. Following a similar approach as Hansen, the radius (R) of this roll-over
shape was determined by performing a regression of the COPS position over the loading angles
with the equation of the lower half of a circle(Hansen et al., 2000).
Y X
22
0 s 0ySCOP R COP x (5.2.3)
This is shown by Eq.(5.2.3) where the COPSX and COPSY
respectively represent the center of
pressure, and x0 and y0 represent the center of the circle in the local X and Y-axis. A plot of the
roll-over shape over fifteen angles is shown in Figure 49. In comparison to the roll-over profiles
from literature, shown in Figure 5, it was noted that there was a gap in the distribution of COPS
data points. This caused by the split geometry of the toe and heel in the FEA model. The position
of the COPS data points, and the curvature of the roll-over shape are directly influenced by the
compliance of the heel and the toe.
77
Figure 49: FEA Roll over shape
To reduce computational time in the parametric design, it was determined that only four
angles, -15°, 0°, 10° and 20°, would be sufficient to determining approximate roll-over shape in
the parametric design. It was assumed that the goal driven roll-over radius of the parametric
model should be 30% of the leg length. Using Drillis’ anthropomorphic model of the human
body shown in Figure 50, the roll-over radius of the model foot was approximated 26.52 cm
(Drillis et al., 1966).
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
-150 -100 -50 0 50 100 150 200
Y-ax
is C
ente
r o
f P
ress
ure
, CO
Py
(mm
)
X-axis Center of Pressure, COPx (mm)
FEA Model
Circle Eq.
10.00°
25.00°
30.00°
35.00°
-19.253°
0.00°
20.00° 15.00° 10.00° 5.00°
30.00° 25.00°
35.00°
-15.00° -10.00° -5.00°
78
Figure 50: Anthropomorphic diagram (Drillis et al., 1966)
Additionally, following Adamczyk’s study it was decided that the center of roll-over shape (X0)
should be anteriorly located 7.6 cm from the tibia (Adamczyk et al., 2006). However, a trade-off
plot shown in Figure 51, revealed that with the current dimensional constraints on this 2D design
model it is unfeasible to achieve the desired radius and center position of the roll-over shape.
79
Figure 51: Roll-over trade-off plot of center position (X0) and radius (R) for 2D model
Nevertheless, it is observed that there are a number of candidate design points that are better
performing under the design criteria of minimizing the weight and peak stress in the design, and
maximizing the elastic energy in the toe. These design points are shown in Figure 51 by the blue
colored Pareto frontier. This is additionally demonstrated in a plot of the maximum Von Mises
stress over the four angles versus the design mass shown in Figure 52. Here the density of the
carbon fiber was assumed to be 1.6 g/cm3.
80
Figure 52: Trade off plot of maximum Von Mises stress versus mass in 2D parametric analysis
Here it is observed that the design point with the lowest mass and minimal stress appears in the
lower left-hand corner. Interestingly, it is noted that many successful marketed carbon fiber
prosthetic feet, such as the Renegade AT, are also designed with a weight of 0.515 kg (Freedom-
Innovations, 2014).
5.3 Parametric 3D Model
A 3D parametric model of the transversely isotropic foot was generated in order to
evaluate the response of the Renegade foot during experimental testing and to examine the effect
of the foot alignment during compressive loading. This was accomplished by selecting the three
thickness variables from in the 2D parametric model and shown in Figure 47, as well as three
81
dimensions of the foot’s profile depth (𝐷) along the toe and keel of the foot shown in red in
Figure 53 where the pylon width is considered to be constant.
Figure 53: Depth (𝐷) dimensions of the foot that were selected for parametric analysis
Using a central composite design of experiments in the Ansys FEA software at total of 45
geometries were generated to examine the response at three levels for each parametric
dimension. The thickness of the heel (𝑇𝐻) was varied between 0.40-0.60 cm. The thickness of the
keel component was varied between 0.75-1.25 cm at the pylon face (𝑇𝐾1) and between 0.40-0.60
cm at the toe (𝑇𝐾2). Similarly, the profile depth in the two components was also varied at the two
major inflection points in the design. At the keel joint the profile depth of the heel (𝐷𝐻1) was
varied between 2.0-3.6 cm and the depth of the keel (𝐷𝐾) was varied between 3.5-4.72 cm, and at
the heel joint the profile depth of the heel component (𝐷𝐻1) was varied between 4.5-5.5 cm. To
prevent rendering errors from occurring during the geometry update, several constrained
82
dimensions (Figure 53Error! Reference source not found.) were added to the CAD drawing to
ensure that the model was fully define. This consisted of restricting foot height to 16 cm, the
length to 23 cm, and placing several additional constraints along the foot profile.
Similar to the 2D parametric analysis, the foot geometry was subjected to static loading at the
angles of -15, 0, 10 and 20 degrees over the ISO 22675 loading curve. By selecting these four
angles, it was possible to develop a response plot of the foot from the heel strike to the push-off.
This was accomplished by writing an APDL script in Ansys to output the 𝐶𝑂𝑃𝑆𝑋 and 𝐶𝑂𝑃𝑆𝑌
of
the foot, and to ensure that each geometry was placed in contact with the compression plate at
each of the roll-over angles. To prevent meshing errors, 10-node tetrahedral elements with
automated meshing used to examine the parametric geometries with an average of 16919 total
elements in each geometry. Additionally, boundary conditions were applied to enforce sliding
contact between the heel and toe with the load plate, and the plyon face was constrained to allow
motion only in the Y-axis, as shown in Figure 54.
83
Figure 54: Boundary conditions used on 3D parametric model with 0° toe out
While the ISO 22675 standard recommends a 7° lateral rotation (𝜏) of the prosthetic foot
to simulate the toe-out during cyclic loading, many experimental studies do not include this
rotation during their investigation (Geil, 2002; Haberman, 2008; Unnthorsson et al., 2008). To
examine the effect of a lateral rotation on the prosthetic foot during loading, the parametric
model was rotated counterclockwise about the tibial axis. This is shown during the heel-strike in
Figure 55a and push-off in Figure 55b.As a result an additional torque was placed on the toe and
heel components during the rollover.
84
a)
b)
Figure 55: FE model of prosthetic foot with 7° lateral rotation about pylon axis
It was observed that the prosthetic foot with the 7° toe out experienced a greater stress
concentration on along the inner and outer surface during compressive loads. A comparison of
the principle stresses on the original geometry is shown in the updated Table 10.
Table 10: Maximum principle stresses due to lateral rotation in original geometry
Transversely Isotropic
𝝉 = 𝟎°
Transversely Isotropic
𝝉 = 𝟕°
Stress 𝜎𝑖 (MPa) at -15° -620.23 -614.76
Stress 𝜎𝑜 (MPa) at -15° 413.82 395.84
Stress 𝜎𝑖 (MPa) at 20° -372.29 -388.23
Stress 𝜎𝑜 (MPa) at 20° 312.99 287.45
Here it can be seen that the foot with 𝜏 = 7° on average experienced an 18% greater stress
magnitude during the loading of the heel and a 10% greater stress magnitude during the
7° 7°
85
compressive loading of the toe. This suggests that the alignment of the prosthetic foot must be
considered during the design phase of the device.
A parametric analysis was conducted on the prosthetic foot with 𝜏 = 0° and 𝜏 = 7°. The
response plot of each parametric analysis was constructed using a genetic aggregation method in
the Ansys Design Explorer to examine the output response variables in each FEA model. These
variables included the maximum tensile stress, compressive stress, and Von Mises stress, the
maximum moment on the ankle joint during roll-over, the strain energy at -15° and 20° loads, the
mass of the foot, and the center of pressure location at each angle. This allowed for a tradeoff
plot to be generated showing the response of the potential design geometries. The tradeoff
response of the principle stresses in the prosthetic foot versus the mass of each design with 𝜏 = 0°
is shown in Figure 56a-b, and the tradeoff response of the foot with 𝜏 = 7° is shown in Figure
59a-b.
86
a)
b)
Figure 56: Trade off plot of tensile a) and compressive stress b) versus mass in 3D parametric
analysis with 0° toe-out
Pareto wt.
Min. 𝜎1 Min. mass
Mass, m (kg)
Ten
sile
Str
ess,
𝜎1 (
MP
a)
Mass, m (kg)
Co
mp
ress
ive
Str
ess,
𝜎3 (
MP
a)
Pareto wt.
Max. 𝜎3 Min. mass
87
a)
b)
Figure 57: Trade off plot of tensile a) and compressive stress b) versus mass in 3D parametric
analysis with 7° toe-out
Pareto wt.
Min. 𝜎1 Min. mass
Mass, m (kg)
Ten
sile
Str
ess,
𝜎1 (
MP
a)
Mass, m (kg)
Co
mp
ress
ive
Str
ess,
𝜎3 (
MP
a)
Pareto wt.
Max. 𝜎3 Min. mass
88
In principle stress response plots shown in Figure 56 and Figure 57, Pareto weights were
added to the feasibility plot such that the mass was minimized and the stress magnitude in
compression and tension were also minimized. As expected, a trend is apparent in the tradeoff
plots for both the principle tensile stress and compressive stress showing that the stress
magnitude can be reduced as the mass of each design is increased. A direct comparison cannot
be made to the Renegade foot with a mass of 0.337 kg because several components including the
metal pylon and overload spring were not included in this FE model. However, an approximation
of the service life can be estimated using Corum’s (Corum, 2001) fatigue model (R=0.1) for 0/90
woven carbon fiber shown in Eq.,
209
T
ln 7.162 10exp
77.47
fN
(1)
Where the maximum tensile stress (𝜎𝑇) in the design can be estimated by substituting in a target
number of cycles to failure (𝑁𝑓). Hence, for a design to withstand a minimum of 106 cycles the
geometry should experience a maximum of 427.96 MPa, which is well within the feasible design
shown in design Figure 56. The compressive fatigue limit 106 cycles for was also approximated
as 85% of the static compressive strength or approximately -406.30 MPa [28]. It can be seen in
Figure 59 that there are also a number of acceptable designs at or below this stress magnitude. It
is also apparent that the most favorable designs along the Pareto front shown in blue.
A response plot of the roll-over shape of the foot under with 𝜏 = 0° and 𝜏 = 7° was
generated from the parametric analysis, respectively shown in Figure 58 and Figure 59.
89
Figure 58: Roll-over tradeoff plot of center position (x0) and radius (R) for 3D model at 0° toe-out
Figure 59: Roll-over tradeoff plot of center position (x0) and radius (R) for 3D model at 7° toe-out
X-axis roll-over center, 𝑥0 (mm)
Ro
ll-o
ver
rad
ius,
𝑅 (
mm
)
Pareto wt.
Min. mass 𝑅 = 282.4 mm
𝑥0 = 76 mm
𝜎1 < 426.96 MPa
𝜎3 > -406 MPa
X-axis roll-over center, 𝑥0 (mm)
Ro
ll-o
ver
rad
ius,
𝑅 (
mm
)
Pareto wt.
Min. mass
𝑅 = 282.4 mm
𝑥0 = 76 mm
𝜎1 < 426.96 MPa
𝜎3 > -406 MPa
90
Pareto weight were added to Figure 58 and Figure 59 such that the x0 was set equal to 7.6 cm and
R equals 28.24 cm. Additionally, a constraint was added to limit the design points to those with a
stress less 427.96 MPa in tension and 406.30 MPa in compression. It was observed that the
desired roll-over radius was well within the scope of the parametric design, however, the changes
to the foot geometry did not allow for the target x0 value to be reached, suggesting that further
design exploration may be needed to achieve this idealized offset value.
A comparison of the work from the vertical force-displacement of the Renegade design at
during heel-strike and push-off with 𝜏 = 0° and 𝜏 = 7° is presented in Figure 60 and Figure 61.
Pareto weights were added to each plot in order to maximize the energy storage capacity,
minimize the mass, limit the compressive and tensile stress, and to attempt to reach the idealized
roll-over shape. There is an apparent tradeoff in the potential designs that are capable of
maximizing the amount of energy that is returned during both the heel-strike and the push-off.
The response of the prosthetic foot at these two critical angles suggest that the toe of the
Renegade foot may be designed to elastically store about three times the energy of its heel.
It was also observed that when the foot is subjected to a compressive load with 𝜏 = 7°, the
foot may be able to store a slightly greater amount of energy than the direct loading (𝜏 = 0°) of
the foot. This is likely due to the additional torque that the pronated foot experiences during
pronated loading shown in Figure 55.
91
Figure 60: Work done by vertical-displacement during heel-strike and push-off at 0° toe-out
Figure 61: Work done by vertical-displacement during heel-strike and push-off at 7° toe-out
Work from vertical displacement, W20° (J)
Pareto wt. Min. mass
𝑅 = 282.4 mm
𝑥0 = 76 mm
𝜎1 < 426.96 MPa
𝜎3 > -406 MPa
Max. W20°
Max. W-15°
Renegade foot
(19.35,9.37)
Work from vertical displacement, W20° (J)
Wo
rk f
rom
ver
tica
l d
isp
lace
men
t, W
-15
° (J
)
Pareto wt. Min. mass
𝑅 = 282.4 mm
𝑥0 = 76 mm
𝜎1 < 426.96 MPa
𝜎3 > -406 MPa
Max. W20°
Max. W-15°
92
The experimental results of the Renegade foot are also shown with the parametric data in
Figure 60; indicated by the red star. During the mechanical loading of the Renegade foot in
Chapter 4, it was observed that the Renegade foot was able to elastically store 9.37 joules of
energy at -15 degrees and 19.35 joules at 20 degrees. It is observed that the heel of the Renegade
provides a greater amount of energy return than many of candidate design points. However, the
parametric analysis suggests that there may be other designs that provide a greater amount of
energy return during push-off.
The parametric feasibility analysis presented in this section had demonstrated that
relatively minor changes in the thickness of a prosthetic design can significantly influence the
roll-over characteristics and the mechanical behavior in the design. These results also suggest
that prosthetic design optimization may lead to the production of more robust devices that can be
developed for the gait pattern of a specific gait pattern.
93
CHAPTER 6: CONCLUSIONS
The development and prescription of a prosthetic foot for a specific amputee is a time
consuming process that is driven by iterative feedback and clinical experience rather than
parametric design. The need to improve this production process has led to the identification of
the essential design parameters that influence the gait pattern of amputee, namely, the component
stiffness, the roll-over shape, the weight, the energy return capacity, and the weight of the device.
However, mechanical testing of these properties and the development process has not yet been
standardized, subsequently their influence on an amputees’ gait remains poorly understood. To
help fill this gap in knowledge, this investigation has conducted a case study on Freedom
Innovations’ Renegade foot and TLM Prosthetics’ TaiLor Made foot to demonstrate that
mechanics of materials approaches can be utilized to evaluate the mechanical response of
prosthetic feet. Additionally, this paper has provided a framework for design optimization
process by conducting a parametric finite element analysis on a representative foot model. A
brief summary of these accomplishments and recommendations for future work are presented in
the following sections.
6.1 Concluding Statements
In order to evaluate the mechanical response of the Renegade and TaiLor Made
prosthetic feet a series of compressive tests involving monotonic loading and force relaxation
were conducted with a universal test machine using a customized loading platform at fifteen
different angles. The compressive loads and angles used during these tests were adapted from the
ISO 22675 standard for the fatigue testing of foot-ankle devices in order to simulate the vertical
94
forces experienced by prosthetic foot during regular walking. In Chapter 4, it was shown that the
both the Renegade and TaiLor Made have an excellent energy return efficiency and storage
capacity during push off at 20 degrees. The Renegade foot which was designed for a 100kg
individual showed energy capacity of 19.50 joules and an energy return efficiency of 92.03%,
whereas the TaiLor Made, which was designed for a 60 kg individual, demonstrated an energy
capacity of 13.54 joules with an energy return efficiency of 83.26% without their foam covers. It
was also observed that the foam cover caused a decrease in the overall efficiency by 3.2 – 5.47%
during the push off phase. Interestingly, the TaiLor Made behaved with greater energy return
efficiency at 0 degrees during loading with its cosmetic cover foam cover. It was also noted that
due the tibial springs in the TaiLor Made design allowed for a greater compliance during the
mid-stance phase when both the heel and toe contacted the load plate.
During the force relaxation testing each foot was compressively loaded and held at a
fixed displacement for a period of 180 seconds in order to capture the viscous behavior of the
component. A regression model using the Norton-Bailey power law for constitutive creep
models was generated for this data. It was shown that the cosmetic foam cover predominated the
viscous behavior of both feet and had a much greater influence on the heel of the TaiLor Made
foot. The experimental results from this work are to be presented in journal publication entitled
the “Mechanical Characterization of Prosthetic Feet and Shell Covers Using a Force Loading
Apparatus”.
In Chapter 5, finite element analyses (FEA) were conducted to approximate the material
properties and stress in the Renegade foot during loading based on the force deflection response
and digital image correlation in Chapter 3. This material approximation was then used in a series
of parametric models to generate a feasibility plot for the mechanical response of potential
95
designs. The use of the parametric optimization and mechanical testing approaches presented in
this thesis could allow for the production of prosthetics devices with a well understood
mechanical response and potentially lead to improved understanding of the influence a prosthetic
device has on the gait pattern of a lower limb amputee. The findings and parametric outline in
this study are to be presented in a publication entitled “Parametric design of mechanical response
in prosthetic feet”.
6.2 Recommendations for future work
While this work has successfully applied mechanics of materials and numerical
optimization techniques to development of lower limb prosthetics, many of these approaches
require further validation through the testing of additional new quality existing prosthetic feet,
the production of a parametrically optimized design, and ultimately biomechanical gait analysis.
To further address the limitations of this methodologies presented in this study, the following
recommendations are for future investigations are provided.
Conduct digital image correlation (DIC) tests as a validation tool in the prosthetic
design process in conjunction with finite element analysis (FEA). The digital image
correlation technique was shown that it can be successfully used to measure the surface
strain along the cross-section of a prosthetic device. However, as the FEA analysis
showed the maximum strain in the design did not occur at the speckle painted surface.
Thus, it is recommended that DIC be used to validate the stress in a FEA model when the
material properties are known.
Development of a frictionless surface for the prosthetic loading plate. During the
course of the compressive testing slippage of the foot without the cosmetic cover was
96
occasionally noticed in the first one to two loading cycle of the five cycle preload. While
these frictional forces were considered to be negligible,
Conduct transient dynamic finite element analysis for walking gait pattern. To
further extend the design optimization methods presented in this paper, it is also
recommended that an explicit dynamic simulation be incorporated in the parametric
analysis. A transient model could provide additional insight into the role a prosthetic
device has on the gait pattern of an amputee and the reaction forces acting on a patients
joints.
97
APPENDIX A: CODES
98
List of ANSYS APDL CODES
Code 1: Offset APDL Command Snippet ..................................................................................... 99
Code 2: Output and Calculations APDL Command Snippet ...................................................... 102
99
Code 1: Offset APDL Command Snippet
!===============================================================
3/27/2016
!
!MOMRG-UCF-MAE
!FOOT RE-POSITION & BOUNDARY CONDITIONS
!KEVIN C. SMITH
!FUNCTIONS:
! - Code offsets foot to touch plate if gap is greater than tolerance
! - Calcualtes the angle theta based on Ankle and Ankle2
! - Fixes lowest point on heel and toe at selecct angles to single node to surfacce contact errors
! Required named Selections in Workbench:
! Ankle
! Ankle2
! TOE_NODES
! HEEL_NODES
! SURF_NODES
/prep7
CMSEL,S,Toe_NODES,NODE
CMSEL,A,HEEL_NODES,NODE
CM, Foot_NODE, NODE
CMSEL, S, Foot_NODE, NODE
*vget, FN_list,NODE,all,NLIST
*GET, FN_count, NODE, 0, COUNT,
*GET, FN_YMIN, NODE, , MNLOC, Y
CMSEL,S,SURF_NODES,NODE
!*GET, PLATE_NODE_NUM, NODE, 0, NUM, MAX
!*GET, Y_SURF, NODE, PLATE_NODE_NUM, LOC, Y
*GET, Y_SURF, NODE, , MXLOC, Y
NSEL, R, LOC, Y, Y_SURF
esln
*get, plate_ELM_num, elem, , num, min
*GET, plate_ELE_TYPE,ELEM,plate_ELM_num ,attr,type
CM, SURF_NODES, NODE
D,All,UX,0
D,All,UY,0
D,All,UZ,0
allsel
100
Offset = Y_SURF - FN_YMIN
AbsOffset = abs(offset)
Tolerance = 1E-6
!offset nodes & elements if not near surface
*IF,AbsOffset,GT,Tolerance,THEN
CMSEL, S, Foot_NODE, NODE
ESLN
NSLE ! grabs the weak spring nodes attached to foot
NGEN,2,0,ALL,,,0,offset,0
allsel
*ENDIF
CMSEL, S, Ankle, NODE
*GET, Ankle_NUM, NODE, 0, NUM, MAX
*GET, AY, NODE, Ankle_NUM, LOC, Y,
*GET, AX, NODE, Ankle_NUM, LOC, X,
CMSEL, S, Ankle2, NODE
*GET, Ankle_NUM2, NODE, 0, NUM, MAX
*GET, AY2, NODE, Ankle_NUM2, LOC, Y,
*GET, AX2, NODE, Ankle_NUM2, LOC, X,
allsel
CMSEL,S,TOE_NODES,NODE
*GET, TOE_ND_Y, NODE, , MXLOC, Y
NSEL,R,LOC, Y,TOE_ND_Y
ESLN
*get, TOE_ELM_num, elem, , num, min
*GET, TOE_ELE_TYPE,ELEM,TOE_ELM_num ,attr,type
CMSEL,S,HEEL_NODES,NODE
*GET, HEEL_ND_Y, NODE, , MXLOC, Y
NSEL,R,LOC, Y,HEEL_ND_Y
ESLN
*get, HEEL_ELM_num, elem, , num, min
*GET, HEEL_ELE_TYPE,ELEM,HEEL_ELM_num ,attr,type
pi = acos(-1)
theta = atan( (AX-AX2) / (AY-AY2) )*180/pi !CMSEL uses intial positions of
pylon nodes used find theta
101
CMSEL,S,HEEL_NODES,NODE
*GET, HEEL_MIN_Y, NODE, , MNLOC, Y
*IF,theta,LT,0,THEN
!CMSEL, S, Heel_pin, NODE
!ESEL,s,type,,HEEL_ELE_TYPE
!nsle
CMSEL,S,HEEL_NODES,NODE
NSEL,r, loc, y, HEEL_MIN_Y
*get,selmin,node,0,num,min,
nsel,s,node,,selmin
D,All,UY,0
allsel
*ENDIF
CMSEL,S,Toe_NODES,NODE
*GET, Toe_MIN_Y, NODE, , MNLOC, Y
*IF,theta,GE,14,THEN
!CMSEL, S, Toe_pin, NODE
!ESEL,s,type,,TOE_ELE_TYPE
!nsle
CMSEL,S,TOE_NODES,NODE
NSEL,r, loc, y, Toe_MIN_Y
*get,selmin,node,0,num,min,
nsel,s,node,,selmin
D,All,UY,0
*ENDIF
allsel
allsel
/solu
102
Code 2: Output and Calculations APDL Command Snippet
!===============================================================
3/27/2016
!
!MOMRG-UCF-MAE
!Center of Pressure calculations and output
!KEVIN C. SMITH
! FUNCTIONS:
! - calculates the center of pressure (COP) based on the weighted average of the contact forces of
the elements underneath the contact elements
! - COP is calculated in a local coordinate system based on theta and the UY of the foot
! - returns the COPx, COPy, and strain energy to workbench
! - "Rollover" outputs: theta (deg), COP_sx (mm), COP_sy (mm), strain work (J), UY (mm),
smax (MPa)
! - "RY_SURF" outputs contact forces : Node, X-loc',FY
! - "Press Nodes" outputs list of conta nodes
! Note that adjustments may need to be made for contact element type if mesh order is changed
ALLSEL
/post1
SET, first, , , , , , ,
*GET,LSTSET, ACTIVE, 0, SET, NSET ! NUMBER OF SETS
*dim,RY_FSUM_FIXED,array,LSTSET
t=LSTSET ! comment this line for static solvers
!*DO,t,1,LSTSET,1 ! loop for transient
SET,,,,,,,t !Defines the
data set to be read from the results file.
FSUM, , ! SUM
COMMAND OPTIONS
allsel
! FIXED NODES reaction force
103
ALLSEL
esel, s, type, ,plate_ELE_TYPE, !
SELECT CONTACT ELEMENT TYPE 3,4
NSLE !
SELECT NODES ATTACHED TO SELECTED ELEMENTS
*GET, LOAD_SUM, FSUM, 0, ITEM, FY, ! SUM NODAL FORCES
RY_FSUM_FIXED(t) = LOAD_SUM !
PLACES LOAD_SUM INTO 'force' ARRAY
!------------------
CMSEL,S,SURF_NODES,NODE
*GET, PLATE_NODE_NUM, NODE, 0, NUM, MAX
*GET, Y_SURF, NODE, PLATE_NODE_NUM, LOC, Y
ESEL,S,TYPE,,4,9,1 ! Element type 4-9 represents contact type
NSLE,S
ESel, S, type, ,plate_ELE_TYPE
NSLE, U
ESLN
ESEL, U, Type,,4,9,1
ESEL, U, Type,,11
!Nforce,
*GET, PRESS_COUNT, NODE, 0, COUNT,
*GET, PLATE_SUM, FSUM, 0, ITEM, FY,
*dim,PRESS_LIST,array,PRESS_COUNT,3
*dim,nnum,array,PRESS_COUNT,
*vget, nnum,NODE,all,NLIST !get a list of the selected nodes
RY_MAX = 0
RY_MAX_X = 0
SUM = 0
WT_SUM = 0
! Cycles through all the surface nodes
! places node num, x-location, and force into an array
! determines location of max force, and the weighted average of the force
! Note that max force is less realistic than the weighted average of the location
*DO,i,1,PRESS_COUNT,1
allsel
104
*GET, X_i, NODE, nnum(i), LOC, X,
*GET,X_f, NODE, nnum(i), U, X, !*GET,X_f, NODE, 180, U,
X,
Press_X = X_i + X_f
!*GET, Press_load, NODE, nnum(i), RF, FY,
nsel,s,node, , nnum(i)
ESLN
!FSUM,cont
FSUM,
*GET, signed_load, FSUM, 0, ITEM, FY, ! returns total forces, not
nodal
!Press_load = abs(signed_load)
Press_load = signed_load
PRESS_LIST(i,1) = nnum(i)
PRESS_LIST(i,2) = Press_X
PRESS_LIST(i,3) = Press_load
! used to determine weighted average of x-location
SUM = SUM + Press_load
WT_SUM = WT_SUM + Press_load*Press_X
! max force location and value
*IF,RY_MAX,LT,Press_load,THEN
RY_MAX = Press_load
RY_MAX_X = Press_X
*ENDIF
*ENDDO
!*ENDDO ! loop for transient
!------------------------------------------------------
! Determine COP location from new coordinate system at pylon
CMSEL, S, Ankle, NODE
*GET, Ankle_NUM, NODE, 0, NUM, MAX
*GET, AY, NODE, Ankle_NUM, LOC, Y,
*GET, AX, NODE, Ankle_NUM, LOC, X,
*get, A_UY, node, Ankle_NUM, U, Y
105
X_AVG1 = WT_SUM/SUM !nodal reaction forces
/PREP7
k, 101, X_AVG1, Y_Surf
Local, 11, 0, AX, AY+A_UY, , -theta, ! local coordinate system at ankle node on
pylon
CSYS, 11
k,102,0,1,
k,103,0,10,
L,102,103
CSYS, 0
/POST1
SET,,,,,,,t
CSYS, 11
*get, COP_sx_k, KP, 101, LOC,X
*get, COP_sy_k, KP, 101, LOC,y
CSYS, 0
!------------------------------------------------------
! Total Strain energy
allsel
etab, Work, SENE
ssum
*get, total_W, ssum,,item, Work
nsort,s,eqv ! Nodal stress maximum von mises
*get,smax,sort,,max
!*get, runtime, ACTIVE, 0, TIME, WALL ! clock
allsel
!-------- WORKBENCH PARAMETRIC VARIABLES -----------
my_COP_sx = COP_sx_K
my_COP_sy = COP_sy_K
my_total_W = total_W/1000 ! converts mJ to J
!------------- OPTIONAL FILE OUTPUTS -----------------
!PRRSOL, FY
! *CFOPEN, RY_SURF,txt ! DEFAULT OUTPUT IS IN RESULTS FOLDER
! *VWRITE,'Node', 'X-loc','FY'
! %8c %8C%8C
106
! *VWRITE , PRESS_LIST(1,1),PRESS_LIST(1,2),PRESS_LIST(1,3) ! WRITES
ARRAY
! %14.6G%14.6G%14.6G
! ! node number, X_location, RY
! *CFCLOS
! *CFOPEN, Press Nodes,txt ! DEFAULT OUTPUT IS IN RESULTS FOLDER
! *VWRITE , nnum(1) ! WRITES ARRAY
! %14.6G
! ! node number, X_location, RY
! *CFCLOS
! *CFOPEN, Ankle_UY,txt ! DEFAULT OUTPUT IS IN RESULTS FOLDER
! *VWRITE,'UY', A_UY
! %11C %14.6G
! *CFCLOS
! !Variables output for troubleshooting
! *CFOPEN, COP_info,txt ! DEFAULT OUTPUT IS IN RESULTS FOLDER
! *VWRITE,'X_AVG1', X_AVG1
! %11C %14.6G
! *VWRITE,'PRESS_SUM', PLATE_SUM
! %11C %14.6G
! *VWRITE,'SUM', SUM
! %11C %14.6G
! *VWRITE,'COP_sx_K', COP_sx_K
! %11C %14.6G
! *VWRITE,'COP_sy_K', COP_sy_K
! %11C %14.6G
! *CFCLOS
! *CFOPEN, C:\Users\Kevin\Desktop\Output\RollOver,txt,,append
! ! theta (deg), COP_sx (mm), COP_sy (mm), strain work (J), UY (mm), smax (MPa)
! *VWRITE, Theta, my_COP_sx, my_COP_sy, X_AVG1, Y_Surf, my_total_W, A_UY, smax
! %14.6G,%14.6G,%14.6G,%14.6G,%14.6G,%14.6G,%14.6G,%14.6G
! *CFCLOS
107
APPENDIX B: EXPERIMENTAL DATA
108
List of Experimental Data
Experimental Data for Renegade Foot with Cosmetic Cover ..................................................... 109
Experimental Data for Renegade Foot without Cosmetic Cover ................................................ 110
Experimental Data for TaiLor Made Foot with Cosmetic Cover ............................................... 111
Experimental Data for TaiLor Made foot without Cosmetic Cover ........................................... 112
109
Experimental Data for Renegade Foot with Cosmetic Cover
Time (ms) Angle, θ (deg) Peak Load (N) Load Rate (N/s) Stiffness (N/mm) Displacement (mm) Energy (J) Efficiency, η (%)
0.00 -20.00 - - - - - -
53.56 -19.25 634.22 147.33 82.45 15.01 2.84 90.24
101.22 -17.50 1109.04 127.66 102.30 18.97 6.87 93.20
144.43 -15.00 1274.63 122.35 114.58 19.20 8.17 93.35
206.49 -10.00 1121.25 119.94 126.76 15.04 5.84 93.35
255.81 -5.00 920.30 129.07 133.05 11.15 3.70 93.54
299.16 0.00 845.19 130.80 174.06 9.93 2.40 85.24
339.17 5.00 906.39 120.07 68.71 21.08 6.68 86.66
377.22 10.00 1052.51 120.33 65.60 25.07 9.94 88.54
414.20 15.00 1208.33 124.91 59.94 33.05 14.76 90.61
450.73 20.00 1277.33 123.45 51.75 41.81 19.42 92.03
487.32 25.00 1176.87 125.33 42.71 47.60 20.27 93.62
524.38 30.00 875.97 125.35 29.81 46.25 15.62 95.17
562.06 35.00 427.76 104.11 18.94 32.56 5.23 95.80
601.32 40.00 - - - - - -
110
Experimental Data for Renegade Foot without Cosmetic Cover
Time (ms) Angle, θ (deg) Peak Load (N) Load Rate (N/s) Stiffness (N/mm) Displacement (mm) Energy (J) Efficiency, η (%)
144.43 -15.00 1276.96 111.77 100.48 15.88 9.37 98.96
299.16 0.00 845.90 132.78 169.46 8.71 2.50 88.17
450.73 20.00 1276.64 115.83 50.36 39.60 19.35 97.50
111
Experimental Data for TaiLor Made Foot with Cosmetic Cover
Time (ms) Angle, θ (deg) Peak Load (N) Load Rate (N/s) Stiffness (N/mm) Displacement (mm) Energy (J) Efficiency, η
0.00 -20.00 0.00 - - - - -
53.56 -19.25 455.68 113.76 27.55 34.19 4.99 74.77
101.22 -17.50 802.89 248.75 45.61 42.52 9.60 73.61
144.43 -15.00 922.05 214.65 53.56 44.48 10.89 73.79
206.49 -10.00 811.03 181.42 47.73 42.21 9.51 74.52
255.81 -5.00 664.27 169.81 40.24 39.19 7.44 74.43
299.16 0.00 610.02 114.15 48.36 32.79 5.25 70.99
339.17 5.00 650.11 151.17 52.06 24.64 4.96 71.05
377.22 10.00 758.17 158.34 48.28 31.42 7.34 73.94
414.20 15.00 871.02 161.06 45.61 37.97 10.20 74.80
450.73 20.00 921.00 170.01 44.96 43.87 12.60 80.02
487.32 25.00 849.12 170.82 34.45 49.81 13.88 84.25
524.38 30.00 631.59 113.08 22.99 50.57 11.10 87.53
562.06 35.00 310.72 102.85 12.66 39.42 4.28 88.62
601.32 40.00 0.00 - - - - -
112
Experimental Data for TaiLor Made foot without Cosmetic Cover
Time (ms) Angle, θ (deg) Peak Load (N) Load Rate (N/s) Stiffness (N/mm) Displacement (mm) Energy (J) Efficiency, η (%)
144.43 -15.00 936.45 272.70 85.37 15.77 6.12 88.46
299.16 0.00 597.47 80.37 43.71 19.71 4.21 59.40
450.73 20.00 919.96 148.17 40.42 44.15 13.54 83.26
113
APPENDIX C: SCHEMATICS OF COMPONENTS
114
Figure 62: Drawing of baseplate to interface with load frames
115
Figure 63: CAD of experimental set-up in MTS 5kN Insight load frame
116
APPENDIX D: TEST MATRIX
117
List of Test Matrices
Renegade Test Matrix ................................................................................................................. 118
TaiLor Made Test Matrix............................................................................................................ 119
118
Renegade Test Matrix
Angle Time (ms) P3 Loads (N) Displacement Rate (mm/s)
-20.00 0.00 0 0.0
-19.25 53.56 630 1.70
-17.50 101.22 1106 1.20
-15.00 144.43 1271 1.00
-10.00 206.49 1118 0.90
-5.00 255.81 916 0.90
0.00 299.16 842 0.70
5.00 339.17 903 1.70
10.00 377.22 1049 1.80
15.00 414.20 1204 2.00
20.00 450.73 1273 2.25
25.00 487.32 1172 2.70
30.00 524.38 872 4.00
34.97 562.06 422 5.94
40.00 601.32 0 0.0
119
TaiLor Made Test Matrix
Angle Time (ms) P3 Loads (N) Displacement Rate (mm/s)
-20.00 0.00 0 0
-19.25 53.56 453 4.0
-17.50 101.22 795 4.0
-15.00 144.43 914 3.0
-10.00 206.49 803 3.0
-5.00 255.81 658 3.0
0.00 299.16 606 2.0
5.00 339.17 649 3.0
10.00 377.22 754 3.0
15.00 414.20 866 3.0
20.00 450.73 915 3.0
25.00 487.32 843 4.0
30.00 524.38 627 4.0
34.97 562.06 304 8.3
40.00 601.32 0 0.0
120
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